NONPARAMETRIC COMPARISON OF SLOPES OF REGRESSION LINES By RAYMOND RICHARD DALEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1984 ■'•-.■= •;<r;->fl'-» To my parents ACKNOWLEDGEMENTS Dr. P.V. Rao suggested the research topic and, as my major professor, provided many hours of consultation with me during the preparation of this dissertation. For this I thank him sincerely. I also thank the other members of my committee, the Department of Statistics, and the staff of the Biostatistics Unit. I am grateful to Dr. Anthony Conti for allowing me flexibility in my work schedule that facil- itated completion of this dissertation. I will not forget Alice Martin for her practical advice and heartfelt support during the various qualifying exams. I thank my close friends for supporting my choice of academic goals v/hile showing me the benefits of a balanced life. I especially thank Christopher Kenward for his encouragement and positive attitude. Finally, I appreciate the love and support of my parents in all areas of m.y life and I am proud to share the attainment of this long-awaited goal with them. ..4^_j»5>„— -fl TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii ABSTRACT V CHAPTER ONE INTRODUCTION 1 1.1 The Basic Problem 1 1 . 2 Literature Review 3 1.3 Objectives and Overview 9 TV70 COMPARING THE SLOPES OF TWO LINES 11 2.1 Introduction 11 2.2 Asymptotic Distributions 16 2.3 Large Sample Inference 32 2.4 Asymptotic Relative Efficiencies ... 35 2.5 Small Sample Inference 68 2.6 Monte Carlo Results 75 THREE COMPARING THE SLOPES OF SEVERAL LINES ... 99 3.1 Introduction 99 3.2 Asymptotic Theory and a Proposed Test 104 3.3 An Exact Test 138 3.4 A Competitor to the Proposed Test . . . 141 3.5 Asymptotic Relative Efficiencies . . . 161 FOUR CONCLUSIONS 171 BIBLIOGRAPHY 175 BIOGRAPHICAL SKETCH 178 vV Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NONPARAMETRIC COMPARISON OF SLOPES OF REGRESSION LINES By Raymond Richard Daley December, 19 84 Chairman: Dr. Pejaver V. Rao Major Department: Statistics Distribution-free confidence intervals and point estimates are defined for the difference of the slope parameters in a linear regression setting with two lines, assuming common regression constants are used for both lines. A statistic, T, is proposed for this setting that is an application of a statistic discussed by Sievers in his paper titled "Weighted Rank Statistics for Simple Linear Regression," appearing on pages 628-631 of The Journal of the American Statistical Association in 1978. The statistic T employs weights, a , that are chosen by the user. If the regression constants are x^ , . . . ,x , then under the weights a = x -x , the null distribution of rs G r T depends on the regression constants. An. associated exact confidence interval for the slope difference can be obtained by calculation of a permutation distribution, requiring use of a computer. Under the weights a = 1 if x -x is ^ rs s r positive, zero otherwise, the null distribution of T is essentially that of Kendall's tau. In this case, an associ- ated exact confidence interval for the slope difference can be calculated using readily available critical values of this distribution. Simulation results indicate the power of a test of parallelism based on T under either of the two sets of weights dominates the power of other available exact proce- dures. Pitman asymptotic relative efficiencies under equal spacing of the regression constants also favor a test based on T over other exact tests. Under unequal spacing of the regression constants, the simulation results suggest use of the weights a = x -x , when feasible. ^ rs s r The method applied in the case of tv/o regression lines is generalized to construct a test for use in comparing the slopes of several regression lines with the slope of a standard line. The proposed test statistic is a quadratic form in a set of statistics, each having the structure of T. The asymptotic relative efficiency of the statistic with respect to a potential competitor is examined. vt' CHAPTER ONE INTRODUCTION o 1.1 The Basic Problem There are many experimental settings where the appro- priate statistical analysis involves comparing the slopes of regression lines. For an example, consider a dilution assay in which two different drugs are being compared to get a measure of relative potency. A linear regression of the response on the chosen dosage levels is constructed for each drug. A fundamental assumption of this type of assay is the assumption that the two regression lines are parallel (Finney, 1964, p. 108). Hence a statistical test of paral- lelism of two regression lines would be desirable in this type of study. In other applications, an estimate of the difference in slopes might be needed. Consider a study comparing the effectiveness of two different fluoride dentifrices in reducing dental decay, where the participants engaged in varying numbers of supervised brushing sessions. The response measured is the change in the number of decayed and filled surfaces over the duration of the study, called the DFS increment. For each fluoride, the linear regression of DFS increment on number of supervised brushings usually results in a negative slope; as the number of supervised brushings increases, the average DFS increment tends to decrease. In this case the dental researcher might be interested in an estimate of the difference between the rate of decrease for the two fluoride groups, that is, an esti- mate of the difference of the slopes of the two regression lines. We have just offered two examples of the basic problem discussed in this dissertation, comparing the slopes of two or more regression lines. In this work we deal with linear regression models in the designed experiment, where the levels of the independent variable are determined by the experimenter, such as dose in a dilution assay. The methods of inference we propose assume the levels of the independent variable are chosen to be the same for each regression line, as is frequently the case in a designed experiment. Thus, in the dilution assay example, the chosen dosage levels for both drugs would be the same. Before proceeding further, we might mention that the problem of comparison of slopes does not suffer from lack of attention. As will be clear from the literature review in Section 1.2, there are many methods for comparing the slopes of two or more regression lines. The classical methods based on least squares theory are exact when the underlying distribution is normal and these methods are asymptotically distribution-free over the class of distributions having finite positive variance. Nonparametric methods exist which are either distribution-free or asymptotically distribution- free over a larger class of underlying error distributions, including heavily tailed ones such as the Cauchy distribu- tion under which the least squares methods perform poorly. The nonparametric, asymptotically distribution-free methods usually have good efficiency properties. Unfortunately, not only is their performance for small sample sizes suspect, but these methods are hard to implement in practice. In this work, we concentrate on the problem of developing better distribution-free methods of inference about slope differences in situations where the experimenter has control over the selection of the levels of the independent vari- able. 1.2 Literature Review In this section we will review articles in the litera- ture discussing nonparametric analysis of linear regression models. We focus on works that are pertinent to the topic of this dissertation, comparing the slopes of two or more regression lines. Often a nonparametric procedure designed to make inference about the slope of a single line may be adapted to the case of several lines. In Chapter Two we will indicate a simple method of adapting a test designed for the single line setting to the two line setting, when the levels of the independent variable for both lines are chosen to be the same. For these reasons we include a discussion of the techniques applicable in the simple (single line) linear regression setting. To begin the discussion, consider the simple linear regression model, Yj = a + exj + Ej, (1.2.1) j = 1, .,., N. The y's are the observed responses, a and B are unknown parameters, the x's are the known regression constants (levels of the independent variable), and the E's are unobservable random errors. In this setting, nonpara- metric tests based on rank statistics have been examined by many authors (Hoeffding, 1951; Terry, 1952). Hajek (1962) discussed the general properties of tests based on the linear rank statistics. N V = E (X -x)<j,(R ), (1.2.2) j=l J 3 where R^ is the rank of Y^ among Y^ , Y^ , ..., Y^^, 9 is a score function which transforms the ranks to scores in an N appropriate way, and x = Z x./N. Hajek discussed j=l ^ asymptotic properties of these tests which depend on the chosen score function. Good discussions of properties of the class of linear rank statistics (1.2.2) are found in Hajek and Sidak (1967) and Randies and Wolfe (1979) . Tests of the hypothesis B = based on the linear rank statistics (1.2.2) can be used to derive estimates of g in the model (1.2.1) by the application of a general technique developed by Hodges and Lehmann (1963) . Since its appear- ance in the literature, this technique has been applied to estimation problems in a wide variety of settings. Because Hodges-Lehmann estimates are used frequently in this disser- tation, we now give a brief example describing this estima- tion technique. Suppose we assiime the simple linear model (1.2.1) but with zero intercept, Yj = exj + E., (1.2.3) j = 1, ..., N, and we use the statistic (the numerator of the Pearson product moment correlation) N Q = Z (X -x)Y. (1.2.4) j=l J -* to test B = against the alternative 6 5^ . Suppose further that the errors E , j = 1, ..., n, are symmetrically distributed about zero. Then N Q(6) = Z (x.-x) (Y.-Bx.) j=l J 3 3 n'n'»)f,>S«?-»« N = Z (x^-x)E^ (1.2.5) has median zero. Thus a desirable property for an estimate B of 6 is that Q(3) be as near as possible to the median of the distribution of Q(3), that is, as near as possible to zero. Then the Hodges-Lehmann estimate of B based on Q is the value of B such that Q(B) = inf |q(B) |. Of course this B example is in a special setting, but it does provide the basic motivation for a Hodges-Lehmann estimate. Adichie (1967) applied the technique of Hodges and Lehmann to derive estimates of the slope and intercept in the simple linear regression setting (1.2.1) using linear rank statistics of the form (1.2.2). Jureckova (1971) generalized these estimates to the multiple regression setting. An undesirable feature of the methods of Adichie and Jureckova is that calculation of the estimates and confidence intervals requires iterative procedures. Sen (1968) gave a complete development of an estimate of slope in the simple linear regression setting (1.2.1), that had earlier been suggested by Theil (1950) . The estimate is the median of the slopes Y - Y ^ . x< x^ , (1.2.6) v -, X r s which is easy to calculate and intuitively appealing. The Theil-Sen estimate can be derived by applying the Hodges- Lehmann technique to the test statistic Z2 sgn(x -X )sgn(Y -Y ) , (1.2.7) r<s ^ ^ ^ ^ where sgn (u) = 1, 0, -1 as u is greater than, equal to, or less than zero. The statistic (1.2.7) is the numerator of a statistic known as Kendall's (1955) tau, a measure of association between the pairs (x., Y.). Since the null (3=0) distribution of Kendall's tau is distribution-free and has been tabulated, and the statistic (1.2.7) has an easily invertible form, exact small-sample confidence intervals for 6 are readily constructed using the Hodges-Lehmann technique applied to the method of Theil and Sen; no iterative proce- dure is necessary. Scholz (1977) and Sievers (1978) extended the method of Theil and Sen by using a weighted Kendall's tau. The definition of a statistic equivalent to the one defined by Sievers will be given in Chapter Two. Sievers also showed how to construct confidence intervals and point estimates for the slope 3 . Note that although Scholz ' s work appeared prior to Sievers, the Scholz reference is only an abstract of an unpublished paper and hence we hereafter refer to this technique as the Sievers-Scholz approach. Further discussion of the Sievers-Scholz approach will be given in Chapter Two . Having considered nonparametric methods of inference about the slope in the simple linear regression setting, we now consider those methods applicable to the multiple regression setting. Recall the method of Jureckova (1971) , which was an extension of Adichie ' s (1967) results to the multiple regression case. A test of the parallelism of k >_ 2 lines using Jureckova 's method requires the computa- tion of k individual Hodges-Lehmann estimates, each calcu- lated by an iterative technique. Sen (1969) and Adichie (1974) specify tests of parallelism of k ^ 2 lines that require only a single Hodges-Lehmann estimate of the overall slope, arrived at iteratively. The Sen and Adichie method has good efficiency properties. However neither Sen and Adichie 's approach, nor the technique of Jureckova, provides exact confidence intervals for the individual slopes. Jaeckel (1972) suggested estimating regression param- eters by minimizing a chosen dispersion function of the residuals. He showed that for a certain class of dispersion functions his estimates are asymptotically equivalent to Jureckova' s, but easier to compute. However, Jaeckel 's estimates are also only asymptotically distribution-free and require iterative computations. Of the methods reviewed, only the Theil-Sen and Sievers-Scholz statistics, applicable in the simple linear regression setting, have readily invertible forms enabling easy computation of exact distribution-free confidence intervals for the slope parameter. In the two line setting. .^i-,* "jy >cm M vfj-m^i^ f ^- HM— wwFmp ' ■■ " i f m i *y^f\\»r'» -^ >f , »■ -^ «Tr* tv-='*r"pTr>J>*- ■ two exact procedures are found in the literature. The first, due to Hollander (1970) , is based on a statistic in the form of the Wilcoxon signed rank statistic. The second exact procedure, due to Rao and Gore (1981) , is applicable when the regression constants of the two lines are equally spaced. The statistic of this second procedure takes the form of a Wilcoxon-Mann-Whitney two-sample statistic. These two procedures are used to test the null hypothesis that the two regression lines are parallel. The null distributions of the two statistics are distribution-free, and exact distribution-free confidence interals for the difference in slopes can be readily computed by applying the Hodges- Lehmann technique. More discussion about the Hollander and the Rao-Gore methods will be given in Chapter Two. 1.3 Objectives and Overview Having reviewed the literature regarding nonparametric comparisons of slopes of regression lines, we now state concisely the two objectives of this dissertation. The first objective is to develop efficient, exact nonparametric methods for comparing the slopes of two regression lines when the researcher has control over the choice of the levels of the independent variable. The methods we propose will enable construction of exact distribution-free confi- dence intervals for the difference between the two slopes. The exact techniques of Hollander (1970) and Rao and Gore (1981) , discussed briefly in the previous section, will be 10 direct competitors of the methods we suggest. A comparison of these three techniques in Chapter Two using their Pitman asymptotic relative efficiencies and a simulation study will establish the superiority of the new methods whenever they ' are appropriate. The second objective of this work is to generalize the new methods suggested in Chapter Two to the setting where the slopes of several regression lines are being compared. In Chapter Three we will extend these new methods to the multiple line case when the purpose is to compare the slope of one of k lines, considered a standard or control, to the slopes of the k-1 other lines. A comparison of our proposed test to a modification of Sen's (1969) test will show that our proposed test, in- addition to allowing an exact distribution- free test for small samples not available with Sen's approach, is almost as efficient as the modification of Sen's test when the sample size is large and the error distribution is not too heavily tailed. CHAPTER TWO COMPARING THE SLOPES OF TWO LINES 2.1 Introduction In this chapter we examine the case of two regression lines, assuming the same regression constants are used for both lines. In this section we first establish the notation for the linear regression model and the statistic used in this chapter. We then motivate the form of the statistic and give some background concerning its development. A special characterization of the Hodges-Lehmann estimate associated with the statistic is given. We close this section with a brief look at the contents of the rest of Chapter Two. Consider the linear regression model Y. . = a. + 6 .X . + E. . , (2.1.1) i=l,2, j = l,...,N, and x- _< x^ £ . . . £ x . In this model a,, a , B-, and g^ ^^® unknown regression parameters, the x's are known regression constants, and the E's are mutually- independent, unobservable random errors. We are interested here in making inference about the slope difference. 11 12 6-,_2 = B-, - &2' Since the regression constants are assumed the sanie for both lines, we suggest the use of the Sievers-Scholz approach introduced in Section 1.2 when discussing techniques appropriate in the simple linear regression setting, now applied to the differences Z . = Y, Ij considered by Sievers and Scholz has the representation - '^2i' J=lf''wN. A statistic equivalent to the one T*(b) = {1/N)i:z a^^sgn(Z^-Z^-bx^_^) , (2.1.2) r<s s r rs where x = x - x , sgn (u) = -1,0,1 as u is less than, ^ o o X^ equal to, or greater than zero, and a > are arbitrary rs — ■' weights with a = if x = x . ^ rs r s Note that under (2.1.1), the differences Z., j=l, ,N, follow the linear model ^j = ^1-2 "■ ^l-2^j -^ El-2,j' (2.1.3) where a^_2 = a^ - a^, e^_2 = 6^ - 63' ^^^ E^_2^ . = E^. - E_., j=l,...,N. In computing the differences Z., j=l,...,N, we have reduced the two line case to the simple linear regression case, enabling us to apply the approach of Sievers and Scholz to the two line case. Of course the assumption of common regression constants is crucial to this reduction. Let us now motivate the form, of T* (b) by first discussing a special case of (2.1.2) due to Theil and Sen. 13 * * Let Z . (b) = Z. - bx. and consider the pairs (x., Z . (b) ) , j=l,...,N. Under H : g = b, the value of x. has no * effect on the value of Z . (b) = a^_„ + E, _ .. However under H, : Pi_2 > tjf a larger x. will tend to result in a * relatively larger observed Z . (b) = a,_„ + (3^ --b)x. + ^1-2 i* '^^^^ ^ method of constructing a statistic for testing H- : &-,_2 = b is to employ some measure of associa- * tion between the x. and the Z . (b) , j=l,...,N. Theil and Sen used this approach, selecting Kendall's tau as the measure of association: II sgn(x )sgn(Z*(b)-Z*(b)) r<s rs s r [II sgn(x )ZZ sgn(Z^ (b)-Z^ fb))] ^ t<u ^" v<w ^ ZZ sgn(x )sgn{Z -Z -bx ) r<s s r rs * U k ' (2.1.4) * Here N is the number of positive differences x - x , s r' 1 <_ r < s <_ N (keep in mind x^ _< x„ _< . . . _< x ) , N i^) = N(N-l)/2, and we assume no ties among the Z's. The * . and Z . : 3 statistic U-^(b) is Kendall's tau between the x. and Z . (b) , j=l,...,N, for each fixed value of b. Let S = (Z -Z^)/x^^, 1 < r < s < N, denote the slope of the line connecting the observed differences Z and Z . r s Since Z^ - Z^ - bx_^ > if and only if S > b, we see that s r rs rs Uj^, (b) is a function of these slopes. The numerator of U (b) is equal to the difference between the number of slopes. 14 S^ , that exceed b and the number of these slopes which are less than b. Comparing U (b) given by (2.1.4) with T (b) given by (2.1.2), we see the Sievers-Scholz statistic is an extension of the statistic due to Theil and Sen obtained by replacing sgn(x^ ) with a general weighting function a . This allows the slopes determined from points which are farther apart to be given more weight than the slopes determined from closer points. Under fairly general regularity conditions, Sievers has shown the highest efficacy of a test based on his statistic is attained when the slopes S are given weights a = X - X = X . Rewriting T (b) using the optimal weights, a = x , we define J. o XT S T(b) = (1/N)ZZ x^^sgn(Z^-Z^-bx^^) . (2.1.5) r<s s r rs In the remainder of this chapter, we explore the appropriate- ness of T(b) for inference about 6._2- We have seen the Sievers-Scholz statistic is a generalization of the statistic due to Theil and Sen. There is a similar relationship between Hodges-Lehmann estimates of the slope parameter associated with the Theil-Sen and the Sievers-Scholz statistics. A Hodges-Lehmann estimate of ^1-2 ^^^^"^ °^ ^^^ Theil-Sen statistic can be shown to be equal to the median of the set of slopes {S^^: l_<r<s^N, X ^ x } (Sen, 1968) . The corres- ponding Hodges-Lehmann estimate associated with the 15 Sievers-Scholz statistic T(b) is a generalization of the Theil-Sen estimate and can be viewed {Sievers, 1978) as a median of the distribution of a random variable V, where P{V=v} = x^g/x.. if v=S^g, (2.1.6) X.. = EZ X , l_<t<u_<N, Xt^x. Thus the Sievers- t<u t: u Scholz estimate is a weighted median of the slope estimates {S^^: l<r<s<N, X t^x}. rs — — r s In the next section we briefly summarize Sievers' results concerning the asymptotic distribution of the Sievers-Scholz test statistic and estimate, now applied to the two regression line setting. Using these results, we describe large sample inference of e._2 in Section 2.3. Pitman asymptotic relative efficiencies (PAREs) of the Sievers-Scholz procedure with respect to other nonparametric approaches as well as the least squares procedure are given in Section 2.4. These PAREs are derived assuming equally spaced regression constants. In Section 2.5 we propose two exact tests of H^ : Q^_^ = which are easily implemented for small sample sizes. Finally, we close Chapter Two with a Monte Carlo study in Section 2.6. The first part of this study concentrates on comparisons of the Sievers-Scholz asymptotic procedure with others under moderately large samples, while the second deals with exact tests and small samples. Because PARE ' s are available only when the x's are equally spaced, these Monte Carlo simulations emphasize 16 comparisons of the various test procedures under unequally- spaced regression constants. 2.2 Asymptotic Distributions In this section we present some important results concerning the asymptotic distributions of the Sievers- Scholz statistic and estimate. All of these results follow from straightforward modifications of Sievers' (1978) results for the simple linear regression setting. Sievers presents his theorems without proofs, giving reference primarily to the text by Hajek and Sidak (1967) . We repeat these results, now applying them to the two line setting. We indicate references to proofs or supply the basic steps, since some of the results in this section will be needed in Chapter Three when considering several regression lines. As mentioned in the previous section, we are assuming the optimal weights, a = x^ - x^, r < s, and thus we state the results for T(b) in (2.1.5). Note that under these optimal weights, T(b) can be expressed in terms of the ranks of the differences Z . (b) , j=l,...,N, as follows: N * T(b) = (2/N) Z [Rank(Z. (b))x.] - (N+l)5, (2.2.1) j=l ^ 3 where Rank (Z . (b) ) is the rank of Z . (b) among {Z*(b): N r = 1,...,N} and x = Z x./N. Hence, T(b) is a linear rank j = l -• 17 statistic and distributional theory for this class of statistics applies to T (b) . For example, assuming some regularity conditions^ asymptotic normality of T(0) under Hq: e-]^_2 = is immediate (see Theorem 2.2.1). Basic notation . We now give some notation that will be maintained throughout Chapters Two and Three. Let 2 N a = Z (x_.-x)^/N, ^ j = l ^ (2.2.2) ^T,A = ^(^j-^)^/3 = Na2/3, (2.2.3) s-1 s j^;^ s : (2.2.4) X N Z j=r+l (x.-x ) , and so (2.2.5) x.. - X . . = N(x.-x) and (2.2.6) 2 2^-2 ZZ X = EZ (X -X )^ = N Z (x.-x)^, r<s ^^ r<s ^ ^ j=l ^ (2.2.7) by simple algebra. For ease in notation, we assume the underlying error distributions of the two lines are the same, with cumulative distribution function (cdf) F and probability density function (pdf) f. Let E^ , j==l,2,... ■ff U ' HJjJl 'ii ii * w ^ . I 1 111 i »jn i j^j ii un.; ! u t[iiiy*wiwrywwiiig?WM'i"*'^a<>EWI Gil l JI, *M U p 18 designate independent and identically distributed random variables with cdf F and pdf f; G and g will denote, respectively, the cdf and pdf of E^ - E^ . The cdf and pdf °^ ■^l ~ ^2 ~ -^3 ^ ^4 ^^11 ^^ denoted by H and h, respectively. For general pdf g, let I (q) = fq^'{x)dx, provided the integral is finite. The normal distribution with mean y and variance a^ will be designated by the notation N{y,a ). We adopt the 0(O, o{'), °p^'^' °p^*^' ^^'^ '^ notations as in Serfling (1980, p. 1, 8-9) . Let P indicate convergence in probabilitv and ^ indicate convergence in distribution. Finally, let a(b+c) be notation for the interval ( | ab | - | ac | , | ab | + | ac | ) . Conditions. The following conditions are used in the statements of the theorems of this section: N _ 2 2 2 (x.-x) = Na -)■ 00 as N ->■ » j = l ^ , ^ CIO iN . II. max (x.-x) max (x.-x)^ l^J^N -^ l<;i^N - N 2 ^ ° as N ■> Z (X -5)2 Na^ j = l ^ III. G is continuous. IV. G has a square integrable absolutely continuous density g. Condition II, the familiar Noether (1949) condition for regression constants, ensures that no individual regression constant dominates the others in size as N ^ - 19 Theorem 2.2.1 ; Under H^ : ^^_^ = b and condition III, the exact variance of T (b) is [ (1/3 ) + (1/N) ] Na,^ . If ^0" ^1-2 " ^ ^^'^ conditions 1, li, and III hold, then ^^^)/''t,A - N^°'l) (2.2.8) as N ->■ «. Proof; From the definition of the Z.'s (2.1.3), 's - ^ - ^^rs = ^1-2, s - ^1-2, r + <^l-2-^)^rs- <2.2.9) Clearly then, the distribution of T (b) (2.1.5) when 6, = b is the same as the distribution of T(0) when 6-, - = 0. So assuming ^-^_2 = 0, we derive the variance of T(0). Var[T(0)] = Var[(l/N) II x sgn (E, , - E, ^ ) 1 r<s rs ^ 1-2, s 1-2, r'^ * ' r't's''^^"*^ =°''[=9°<=l-2,s-^l-2,r'' =5n (Ei.2,s-=l-2 ,t' ^ " ' r's'u''-^"^" '^°"'=9'"^l-2,s-^l-2,r'' =9" '=1-2 ,u-=l-2 , = ' " = (l/N^) {Cj + Cj + C3 + c^}, (2.2.10) f ¥ * -- ' i--X' ^^g"g^g^|^gg^*W r^i5> "^; :jP!*Q»gJ g '«'**^M M P« *-L» B;i'' J w .i;» i Nai M« Ct . 20 where c^, c^, c^ , and c^ denote the 4 terms in this variance expression. Since E ^ ^ - E is symmetric about zero, it follows that E[sgn(E^_2 ^ - E^_^ ^) ] - o. Also since '2_2 s ~ ^1-2 r ^^ continuous, it follows that 2 sgn (E.|^_2 g - E^_2 j.) = 1 with probability one. Thus ^1 = 11^ x's^^-f^^^ (^l-2,s-^l-2,r)^ 11^ 4s ^f^^^' (^l-2,s-^l-2,r)^ 22 X (2.2.11) r<s ^^ To simplify C2 we first note Gov [sgn (E^.s , g-E^.s ,^) , sgn (Ei_2 ^^-Ei_2 ^^) ] = E[sgn (Ei_2,3-Ei_2,r) sgn (Ei_2 ,^-Ei_2 ,P ] = t^ ^^1-2, s - ^1-2, r ' 0' ^l-2,u " ^1-2, r > '^ + ^ ^^1-2, s - ^1-2, r < 0' ^1-2, u " ^1-2, r < 0>^ - f^ ^^1-2, s - ^1-2, r ^ °' El-2,u - ^1-2, r ^ 0> ^ ^ ^V2,s - ^1-2, r ^ °' ^l-2,u - ^1-2, r > °>^ = [ /[l-F(e)]^dF(e) + /F^(e)dF(e)] - [ /[l-F(e) ]F(e)dF(e) + /F (e) [1-F (e) ] dF (e) ]■ = [(1/3) + (1/3)] - [2(1/6)] 1-2 s "^^ continuous (condition III) and consequently ; 21 - l/3f (2.2.12) and so it follows that '2 ^ \<s</rs-ru ^^^^ ^^^ ^^1-2 ,s-^l-2 ,r) '^^^ ^^1-2 ,u-^l-2 ,r) ^ (1/3) (2 ZZS X X ) rs ru r<s<u (1/3) (I xj ) (using (2.2.5)). (2.2.13) r ^• Similar basic manipulations yield the following simplifications of c- and c . : '3 = (1/3) (Z X h, (2.2.14) r c^ = - (1/3) (2 Z x^^x^^). (2.2.15) Substituting these simplifications into (2.2.10) we have Var [T(0)] = (l/N^) { ZZ x^ r<s ^^ + (1/3) Z (x^^ + x^^ - 2x^_x^^)} = (l/N^) {J:z xJ + (1/3) I (x^ -x )^}. (2.2.1&) r<s ^^ r ^' '^ Using (2.2.6) and (2.2.7) in (2.2.16) yields 22 Var[T(0)] = (1/N^){N E (x.-x)"^ + (1/3)N^ I {x.-S)^} j=l 3 j=i 3 ^ - 9 = Z (x -x)^ { (1/N) + (1/3) } j = l J = [(1/3) + (1/N)] Na^, (2.2.17) o verifying the expression for the exact variance of T (b) under H^ : B = b. If we take k=2 in Theorem 3.2.3 then this theorem states that under H^ : e^_2 = 0' assuming conditions I, II, and III T(0)/a^^^ J N(0,1) (2.2.18) as N ^ 00. From the remarks at the beginning of this proof T(b)/a^^^ has the same limiting distribution under H_ : ^1-2 ^ ^' ^^^ ^° ^^^ proof is complete. Note that 0^ is the asymptotic variance of T(b) under H : 3 ^ = b. Theorem 2.2.2 ; Assume a sequence of alternatives J, ^N* ^1-2 "^ ^1-2^^^ "^ to/(N^a^) to the null hypothesis H- : 2l_2 = 0' where to is a constant. Then under conditions I-IV, '^^°^/''t,A t N(2(3'S)a)I(g),l) (2.2.19) as N ->■ 00 -•»w ajj< > u*j. i i i' m i L i' J»i » iii i i.w i-ii P , III 1 . 1 m m rajnu ,« i M« 23 Proof ; A detailed argument is given, since results obtained here are used in the proof of Theorem 3.2.4 in Chap- ter Three. Let <^i^ = 0)/ (N^o^) . From (2.2.9) and the definition of T(b) (2.1.5), it is clear that the distribution of T(0) under e^_2 = im^ is the sam.e as the distribution of T(-a) ) under ^-^^2 ~ ^' Therefore the proof is complete if we assume &2_-2 ^ ° ^^^ show that T (-cu ) has the desired limiting distribution. V7e first state and prove two lemmas. Lemma 2.2.3 : Assume 3^_ = and conditions I-IV. Then E[T (-03^)70^^^] ^ 2(3')a)I(g) (2.2.20) as N -»■ «>. Proof of lemma: E[T(-a3^)] = (1/N)ZE x^^E [ sgn (Z^-Z^+co^^x^^^) ] . r<s Now E[sgn(Zg-Z^+aj^x^g)] = 1 - 2/G (z-u^x^^) dG (z) . (2.2.21) ^>^>»i^>*v. ->r»V ^>V'>_:.i-^r» 24 Conditions III and IV allow us to write a Taylor's series expansion of G(2-MX) : G(2-cux^) =G(z)-cax a(z-e (z)), j-NXo Mrs Nrs" rs where < |9^^(z)| £ ^N^^rs ' " ^^^""^ max X . -X ^N^rsl = U(x^-V/(n\H < 2 M i£J|N_i N^ a X it follows from condition II that 6 (z) ^ uniformly in r rs -' and s as N -^ ". That is, for all e > 0, there exists N(e) such that N > N(e) => Icoj^x^gl < e for all r < s. (2.2.22) Then by absolute continuity of g (condition IV) , for all 6 > 0, there exists N {&) such that * • N > N (5) => g(2-6^g(z)) e (g(2)±<S), (2.2.23) for all 2. Since g(-) and 6 are nonnegative and g(«) is square integrable (condition IV), multiplying by g(z) and integrating we see * N > N (6) => /g(z-e^g(z))g(z)dz e (I(g)±6). (2.2.24) Substituting the Taylor's series expansion into (2.2.21) we have E[sgn(Z^-Z^+o;j^x^^)] = 2a)^^x^^/g (z-e^^ (z) ) g (z) dz , -.rr- T^v-".;^!,!,. 25 and so E[T(-a.^)/a^^^] = (2a)^/ (Na^ ^^) ) Z Z x^^/g (z-G^^ (z ) ) g (z ) dz X ^ o Assume tu > 0. Using (2.2.24) in the above representation we see N > N (6) => E[T(-aj^)/a^ ^] e {2a)^/ (Na^ ^))IZ x^ (1(g) ±5) ' ' r<s ^ - 2 (2a)/(NG ))(N Z (X -x)^) (I(g)±6) (from (2.2.7)) ' -1=1 J 2(3^)a)(I(g)±6) , and so, N > N* (6/ (2 (3 ') u) ) => E[T(-cj^)/a^^^]e(2(3^)ujI(g)±6) , and (2.2.20) is proved for w > . The proof is similar for for 0) < 0. Lemma 2.2.4 : Assume 6,_2 = and conditions I, II, and III. Then III f^\-*'\^ fm »-»^%r^J'imm'**t r'i:' mm mj m ft .jr 26 as N ^ " . Proof of lemma : Since under &^_^ = 0, E[T(-a)j^) - T(0)] - E[T(-aj^)], we prove (2.2.25) by showing Var[(T(-a)jj)-T(0))/a^^^] -^ (2.2.26) as N ^ ». First, T(-a)j^) - T(0) = ^IZ ^rs^rs ^"n^ ' where E^^it,^^) = sgn (Z^-Z^+oj^x^^) - sgn(Z^-Z^). Assuming co > , 2 -tj„x < Z - Z < otherwise. We will suppress the argument ui^ of H^^ (u^^) . Then, since 'Vrs' "^ ° uniformly in r and s (see (2.2.22)) and the cdf of 2g-Z^ is continuous (condition III) , it follows that 2 E[H^^] = 4P{-aj^x^^<Zg-Z^<0} ^ uniformly in r and s as N > ". Similarly, E[H H , ,] > uniformly in r, s, r' , s' as N > <». So Var(H ), Cov(H ,H ), and other terms of J- o r s r u that form are all uniformly o(l) as N ■» ». Now Var[(T(-a)^)-T(0))/a^^^] (N^aJ ) ^Var[EZ x H ] -' r<s ^^ ^^ lj*t*;:«|«e^»|,-i-«[j> 27 = (N2aJ^^)-l(ZE^x2^Var H^^ + 2 ZEE X X Cov(H ,H ) rs ru rs ru r<s<u + 2 ZEE x^^x, Cov(H ,H^ ) r<t<s ^^ ^^ ^^ t^ + 2 EEE X X Cov(H ,H ) } r<s<u ^^ =^ ^^ ^^ 2 2 -1 " ^^ '^T,A^ (0^+02+03+0^), (2.2.27) where c^, <z^, o^, and c^ denote the 4 terms in this varianoe expression. Since r < s < u implies < x < x , it ^ — rs — ru' follows that C^ < 2 EEE X X |Cov(H ,H ) ~ r<s<u ^^ ^^ ^^ ^^ < 2 EEE x^ ICov(H ,H ) — ru ' rs ru r<s<u = 2EEju-r-l)x2jcov(H^3,H^^)|. Now (u-r-1) <_ N for all 1 £ r < u < N, and c 1 < 2NEE x" |Cov(H ,H ) 2' - ^^^ ru' ' rs' ru' so 28 The two similar terms c_ and c. in (2.2.27) can also be bounded in this v/ay. Combining these bounds with the fact that all variance and covariance terms in (2.2.27) are uniformly o(l) we find Var[((T-(o^)-T(0))/a^^^] < B^, where B^ = (o(l)/N^a^ ^) (6N+1)ZZ x^ ' r<s (3o(l)/N^a^) {6N+l)N^a^ = o (1) , as N -> ", and hence (2.2.26) is proved for u > . The proof is similar for u < . Proof of Theorem 2.2.2 (continued) ; We see by Lemma 2.2.3, Theorem 2.2.1, and Slutzky's theorem (Randies and Wolfe, 1979, p. 424) that under Hq: B^_2 = 0, T(0) + E((T(-a) )) ^^— J N(2(3^)a)I(g) ,1) T,A as N -> «=. Using this result. Lemma 2.2.4 and another application of Slutzky's theorem shovz that under ^0- ^1-2 " °' cffc — To«a »g- iT^ 29 T(-(^N^''''t,A t N(2(3^)wl(g) ,1) as N ^ ". Then by our remarks at the beginning of the proof of Theorem 2.2.2, we are done. Consider the Hodges -Lehmann estimate of 6,_^, say 6, ,, associated with the statistic T(b). If we let ^1_2 = sup{B: T(3) ^0} and i^_^ = inf{B: T(3) < 0}, then we may define Q^_^ = (6^_2+eJ_2) /2 . We now give a theorem concerning the asymptotic distribution of 3t_,. Theorem 2.2.5 ; Under conditions I-IV, ^^^x(^l-2"^l-2^ f N(0,(12l2(g)) 1) (2.2.28) as N ->■ ". Proof : ■^®^ "n ~ '^Z (N'^^^) where w is a constant. Let P„{'} h denote the probability calculated when 3 = 3, and let < designate the standard normal cdf. From Theorem 2.2.2 it follows that lim P {T(0)£0} lim P {(T{0)/a ) - 2(3^)ajl(g) < -2 (3^) cal (g) } ^a j WJT«i » IJ il im i U^j yi . W r-7 <ali 30 (-2(3 )ul(g)), or equivalently , lim P_ {T(C)£0} = * (2 (3^)0)1 (a)) . (2.2.29) By the definition of &^_^ and 3^_2 and using the fact that T(b) is nonincreasing in b, we have ^1-2 ^ ^ "^ ^^^^ ^ ° "^ ^1-2 - ^' (2.2.30) ^1-2 < ^ => T(b) < => bJ_2 ^ b. (2.2.31) Since by condition III the underlying distribution is continuous, it can be shown (using a proof similar to that of Theorem 6.1 of Hodges and Lehmann (1963)) that the distributions of e^_2 and ^2,-2 ^-^^ continuous. Thus (2.2.30) and (2.2.31) imply that Pp ii^ <h} = P {T(b)<0}, (2.2.32) P {8^ <b} =.P {T(b)<0}. (2.2.33) ^1-2 -^ ^ Pi_2 Since B^_2 = (1/2) (6^_2 + 65^_2) and 6^_2 < l^_2' it follows that ^6,_2^^?-2<^> 1 ^B^_2^^1-2<^> 1 ^^_2^^1-2<^>' ! ■»;» " . ii LJ« «M''*qw»Mwwww«»'wM«»«g°w5gT«»»°B wa J i ^ M U-. ■ im i g.'i^m. iii jB.ir^T—'rj 'i mf * m n iiiiiii.i . i i im o— — bm 31 and then substituting (2.2.32) and (2.2.33) into this result we have P {T(b)<0} < P {6 <b} < P. {T(b)<0}. (2.2.34) Because the distribution of T(B) when ^-,_2 = 3 is the same as the distribution of T(0) when 6,_p = (see proof of Theorem 2.2.1) it can be shown using the definition of B that the distribution of (^i_2~^i_2^ under &^_^ is the same A as the distribution of &-,-2 ^^®" ^i_2 "^ ^' Since the dis- A tribution of 3-J.-2 ^^ continuous and the limiting distribu- tion in (2.2.29) is continuous, we apply (2.2.34) to obtain lim P {N'a (6 -3 ) < to} = lim P-^{N^a^6T ^ <cj} lim P {3 < u^/ {N^'o ) } lim P {T(aj/(N'a^) ) < 0} (from (2.2.34)) = lim pQ{T(a3j^) < 0} -J«IJJl.J!'.]JieWW'^WSlJ-.'B i! im.BlMmMW.UM ^i^1l gW<g - «• "■ - ' ■ »i i..r nTOi.iia i. »« j i mi--. ca» 32 = lira P {T(0) _< 0} = §(2(3^) a)I(g)) (from (2.2.29)), (2.2.35) and the result (2.2.28) follows immediately. 2.3 Large Sample Inference Using the results of the previous section, we now proceed to construct tests and confidence intervals for ^l_2- These tests and confidence intervals follow from those presented by Sievers, now applied to the two line setting. Consider a test of H^ : B^_2 = &q against H^ : Q > 6 based on the statistic T(eQ). Of course to test for parallelism of the regression lines we take g. = 0. Large values of T(3q) indicate the alternative H^ holds. For a given < a < 1, let z denote the 100 (1-a) percentile of the N(0,1) distribution. Then from Theorem 2.2.1, the test which rejects H if ■"6o'''='t,a' \ is an approximate level a test. A two-sided test of H '2__2 = is derived by making the usual modifications. Since T (b) (2.1.5) is a nonincreasing step function of b with jumps at the slopes S = (Z -Z ) /x , 1 < r < s < N, a confidence interval for g can be constructed by 33 inverting the two-sided hypothesis test. This interval has endpoints that are properly chosen values from the set of slopes {S : 1 <_ r < s <_ N}. Let J(s) be the cumulative distribution function of a discrete random variable that takes the value S with rs probability x /x. . , where x.. = ZZ x . Then for each b r<s such that b7*S ,l_<r<s_<N, J(b) = (1/2) [l-(NT(b)/x..)] . (2.3.1) If the distribution of the Z ' s is continuous, then P{S^g = b for some l_<r<s_<N} = for each b and (2.3.1) holds with probability one. Therefore if we can determine constants t^ and t^ such that P{t, <T (6) <t2 } = 1 - a, then 1 - a = P{^(l-(Nt2/x..)) < J(e) < J2(l-(Nt^/x..)) } = p{j;;;-^(^(i-(Nt2/x..))) ^ e < jj-(h{i-{i>it^/x. .)))}, (2.3.2) where J~ (u) = inf {s: J(s)>_u} and j"""" (u) = inf {s: J(s)>u} are inverses of J defined appropriately for our purpose. Thus [J^-^(%(l-(Nt2/x..))) , J^^(35(l-(Nt^/x..)) )) (2.3.3) --*-»»^»^«Wi-— .--T^CH';- -.^■.i-a''*>t:i>- »■■"--■ ~r<l.-JL. -..*■■■ T«JC— *».»f*'— ^-^ 34 is a 100 (1-a) percent confidence interval for 6, ^. If V7e use the asymptotic normality of T(3) in Theorem 2.2.1 to determine t^ and t^ we find that ^^~-^ ^'^-^a/2 <N^T,a/2^- • ) ) ' ^'J' ("s+z^/s (Na^^A/2x. . ) ) ) (2.3.4) is an approximate 100 (1-a) percent confidence interval for ^1-2* ^^ ^^^ write this interval more explicitly in terms of the slopes S^^ as [S^'^^s'^) , where S = min{S^g: J (S^^) >%-2 , (Na /2x. . ) , l£r<s<N}, (2.3.5) and S = miniS^^: J {S^^)>h+z^^^{lio^ ^^/2x. .) , l<r<s<N}. (2.3.6) An alternative confidence interval follows directly from the asymptotic normality of &^_^ (Theorem 2.2.5). An approximate 100 (1-a) percent confidence interval for e._ is given by ^1-2 ± 2,/2(2(3V)a^i(g))-l, where 1(g) is a consistent estimate of 1(g). Such estimates have been proposed by Lehmann (1963) and Sen (1966) . 35 2.4 Asymptotic Relative Efficiencies In this section we compare the Sievers-Scholz approach for testing H^ : e^_2 = b with the procedure due to Hollander and the procedure due to Rao and Gore. As noted in Chap- ter One, the Hollander and Rao-Gore procedures include exact confidence intervals for the slope difference 5-,_2' We will see in the next section that our proposed application of the Sievers-Scholz approach also includes exact confidence intervals for &2.-2' ^^^^^- these three exact, nonpararaetric procedures are the primary focus of the asymptotic relative efficiency comparisons presented here. Comparisons of the Sievers-Scholz test with the classical t-test based on least squares theory and Sen's (1969) test are also made. First we show how to construct each of the competing nonparametric . test statistics, along with a brief illustration of the rationale behind each one. We then describe a sequence of sample sizes tending to infinity for the purpose of computing the Pitman asymptotic relative efficiencies (pares) . Finally', we compute these efficiencies and compare their values assuming several different underlying error distributions. Since all three exact, nonparametric procedures can be expressed in terms of basic slope estimates for each line, we first define ^irs = ^"is-^ir^/^rs' ^r " ^^s' (2.4.1) nMMi^«P9f« 36 the estimate of the slope of line i (i=l,2) resulting from the responses at x and x . Recall that in the previous sections we used a similar notation. 3 = (Z -Z )/x , rs s r rs' to designate slope (difference) estimates computed from the differences Z., j=l,...,N. Of course S = S^ - S^ , and 3 rs Irs 2rs the additional subscript indicating line 1 or 2 in the estimates in (2.4.1) should help to avoid any confusion. There will be {^^) = N(N-l)/2 of these estimates associated with each line. The slope estimates of line 1 are naturally independent of those of line 2, but the (2) slope estimates of a single line are not mutually independent. One way of motivating and com.paring the three exact, nonparam.etric procedures is to examine how they utilize these basic slope estimates in forming their test statistics. The Sievers-Scholz Statistic In the two line setting, the Sievers-Scholz statistic is appropriate only when the lines have common regression constants. As before, let x- < x„ < . . . < x„ denote the 1 — 2 — — N regression constants. We can write the Sievers-Scholz statistic T(0) in terms of the slope estimates: T(0) =lEr x^3sgn(s,^.3-S2^^) r<s 37 Examining this representation we see that each line 1 slope estimate is compared v/ith the line 2 slope estim.ate resulting from the observations at the same regression constants. Thus all (p) slope estimates of each line are used, but an estimate from line 1 is compared only with the corresponding estimate from line 2. This results in („) comparisons across lines. Each comparison is weighted by the distance between the x's used in its construction. The Hollander Statistic Unlike the Sievers-Scholz statistic, the Hollander statistic is applicable even when the lines do not share common regression constants. However, use of the Hollander statistic requires a grouping scheme designating N/2 pairs of regression constants for each line. Assume N = 2k. When the x's are equally spaced, that is, when the regression constants for line i are X. T = L. + mc, m = 0, 1, ..., 2k-l, (2.4.2) i = 1, 2, for some constants L^ , L„, c^ , and c_, Hollander's grouping scheme pairs X. ^ with X. ^,, , m = 1, ..., k, (2.4.3) i,m i,m+k i = 1, 2. The first step in Hollander's procedure is to utilize the observations at each pair of x-values to 38 construct N/2 independent slope estimates of the fom (2-4.1) for each line. Hollander notes that under equal spacing (2,4.2) his grouping scheme (2.4.3) minimizes the variance of the slope estimates used among all grouping schemes that produce identically distributed slope estimates. Under unequal spacing of the regression constants, some ambiguity exists as to the choice of a grouping scheme. Hollander suggests devising a scheme that will yield pairs of x-values situated approximately a constant distance apart. Having computed N/2 = k independent estimates of the slope of line 1 and k independent estimates of the slope of line 2, the next step in Hollander's procedure is to randomly pair the slope estimates of line 1 with those of line 2. Let (S-j^^.^, ^2^^) designate one of the k pairs. For each pair the difference of the form S, - S^^ is Irs .2tu calculated. If we label these differences d^ , . . . ,± then 1 k Hollander's statistic is computed as W = ^ ^m^^'^m^' (2.4.4) m=l where R^, is the rank of |d^,| among {|d^|: m=l,...,k}, and 6 (a) = 1 if a > , otherwise; W is the Wilcoxon signed rank statistic computed using the k slope difference estimates as the observations. Writing a slope difference ^•ft w p y M gw pn y 1 1 'im \\ t» m mi-ykm^ imm^Qr»\mo»mv ?9 e'^'* y*l»'SK~mr r 39 estimate d^ in terms of the underlying parameters, we see d has the form where x^^^ = x^^ - x^^, i = 1, 2. Since E^^ and E^^ are independent and identically distributed, it follows that the distribution of E^^' - E^^ is symjr.etric about zero. Clearly then , ^is - ^ir ^2u - ^2t ""irs ^2tu is symmetrically distributed about zero. Thus Hollander's approach does not require the same regression constants or equal spacing since the Wilcoxon distribution will apply under 6^_2 = regardless of the spacing or choice of regression constants. However, if the regression constants are the same and equally spaced, the asymptotic relative efficiency (ARE) to be presented in this section as Result 2.4.2 indicates superiority of the Sievers-Scholz approach. Note that Hollander's approach does not use all available (2) basic slope estimates from each line. Instead a subset of N/2 independent line 1 slope estimates is selected from the {^) possible. A similar set is selected from the line 2 slope estimates. Each member of the first 40 set is compared with only one randomly selected member of the second set, resulting in only N/2 comparisons of slope estimates across lines. This is a sm.aller number of comparisons across lines than the (2) comparisons of the Sievers-Scholz method. Although there are dependencies among the Sievers-Scholz comparisons, the greater number of them would lead us to expect the Sievers-Scholz approach to be superior to Hollander's method when both are applicable. Again, the ARE Result 2,4.2 will confirm this. The Rao-Gore Statistic Let A(B) designate the set of N/2 slope estimates for line 1(2) used in constructing Hollander's statistic. Note that under equal spacing (2.4.2) with c^ = C2 = c and using Hollander's recommended grouping scheme (2.4.3), the distribution of the slope estimates in A differ from those in B only by a shift in location of 3-i_2 "= ^i ~ ^2' E, - E^ q - R + Is Ir ^Irs " ^1 ^ kc c - B + ^2u - ^2t ^2tu ^2 ^ kc Here x. , - x. = kc is the common distance between pairs i,m+k i,m of x's used in forming the slope estimates. Thus, under equal spacing, Rao and Gore proposed the Mann-Whitney- Wilcoxon statistic of the fcirm 41 " -{S.^JeA}{S2^^eB} ^ ^Slrs'^2tu)' (2.4.5) where y{a,h) = 1 if a > b, otherwise. It is clear that the Rao-Gore procedure would apply under any spacing and grouping scheme resulting in the same distance between all pairs of x's used to form slope estimates. Under such spacing, the Rao-Gore procedure eliminates the extraneous randomization needed by Hollander's procedure to pair the slope estimates of the two lines. We see the Rao-Gore procedure uses the same two sets of slope estimates as Hollander's approach. However all 2 possible (N/2) comparisons across lines are made. This leads us to expect that the Rao-Gore procedure will compete favorably with Hollander's method. Comparing the Rao-Gore technique with the Sievers- Scholz approach in the previous intuitive way is not as revealing. The Rao-Gore technique makes all possible comparisons across lines among two relatively small sets of independent slope estimates. The Sievers-Scholz approach makes only pairwise comparisons across lines, but uses all possible slope estimates. The sets of slope comparisons used in the two procedures are such that neither set is a subset of the other. We will see, however, that the ARE Result 2.4.3 indicates superiority of the Sievers-Scholz approach. ■ in.ii i an i i_». i ., 42 The Sen (1969) Statistic Sen (1969) proposed a statistic for testing the parallelism of several regression lines. We introduce his statistic here in the two line setting. Note that Sen's statistic does not require common regression constants for the two lines as required by the Sievers-Scholz statistic. However we will assume common regression constants to ease the notation. Modifications needed to construct Sen's statistic in the more general case of different x's for the two lines will be obvious. Thus we assume the basic linear model (2.1.1) . Let (|) (u) be an absolutely continuous and nondecreasing function of u: < u < 1, and assume that ((> (u) is square integrable over (0,1). Let U^^^ < U^2) < ••• < U^^^ be the order statistics of a sample of size N from a uniform [0,1] distribution. Then define the following scores: Ej = E[MU(j))], (2.4.6) or Ej = <j)(j/N+l), (2.4.7) j = 1, , N. Define 1 ** = / 0(u)du (2.4.8) and ■■I.W .mimu g'ma^'Sf^ 43 2^2 2 A = f ^ (u)du - (<i)*)^, (2.4.9) and consider the statistics V = [ E (x.-x)E ]/(An\ ), (2.4.10) 3=1 -^ ij i = 1, 2, where R^^ is the rank of Y^. among Y., Y.^, ..., ^iN' ^^^ observations of the ith line. Statistics such as (2.4.10) are used in the single line setting to test hypotheses about the slope. The function (j) (u) is called a score function. It is a function applied to the observed ranks of the Y's. The choice of cj) can be made to achieve desirable power properties of a test based on (2.4.10). These properties depend in part on the underlying error distribution. A general discussion of score functions is beyond the scope of this work. We will discuss the score function here only to the extent needed to clearly present Sen's statistic. Assume F, the underlying error cdf of the assumed model (2.1.1) , is absolutely continuous and I*(F) = / [|^]2dF(x) < ». (2.4.11) wp <i"'ji^wj: iT ^p-. g^ BRm a 44 Then we reserve the symbol ^ (u) for the following score function: ,(u) = _ fMZ^iluU , < u < 1. (2.4.12) f(F ^(u)) It can be shown. 1 / 1'(u)du = 0, and 1 2 / 'P (u)du = I*(F) . The score function Y (u) has been shown to have certain desirable properties when applied to the two-sample location problem (Randies and Wolfe, 1979, p. 299). We also define 1 Pi'^r<i>) = [/ 'i'(u)^(u)du]/[A^I* (F)]^, (2 4 13) which can be regarded as a measure of the correlation between the chosen score function <p and the optimal one C?) for the error distribution being considered- The expres- sions p(V,(j)) (2.4.10) and I* (F) (2.4.11) will appear in the development of the statistic based on Sen's (1969) work. 45 We now define Sen's statistic. Let Vidi+bx) (2.4.14) denote the value of V^ (2.4.10) based on Y. + bx , Y. + bx2, ..., Y^^ + bx^. Define V = (V^ + v^)/2. (2.4.15) Assuming H^ : &^_^ = o, let 6* denote the Hodges-Lehmann estimate of the common slope of the two lines based on V. Define Vi = V.(Y.-3*x), (2,4.16) i = 1, 2. Then Sen proposed the statistic . _ 2 .2 ^ ~ -l.^i (2.4.17) to test Hq: B^__2 = against H^ : &^_^ ^ 0. The statistic L is a quadratic form in the V., i = l, 2. We now give an intuitive motivation for the form of L. The statistic V^ is the value of V. based on the observations ^il - ^^-^^1' ^12 - ^*-2 ^iN - e*-N- 46 Under H^ : &^_^ = , the 2 lines are parallel and g* is an appropriate estimator of the corrjr.on slope. In that case, the transformed observations behave essentially as random errors fluctuating about zero. Then L, being the sum of squared random errors with mean zero, has an asymptotic central chi-squared distribution, and a test of H may be based on L using this asymptotic null distribution. Under H^: 6^_2 7^ 0, the estimate B* is not appropriate since the two slopes are not the same. The transformed observations will not, in general, have mean zero. Hence the value of L will be larger than expected under the null, and the use of the null, central chi-squared critical values will tend to lead to rejection of H . PARE Specifics: Alternatives, Regression Co nstants , Sequence of Sample Sizes ' In computing the PAREs, we assume a sequence of alternatives to the null hypothesis H : g = o specified ^y ^N" ^1-2 " ^1-2^^-^ " w/N'a^ as in Theorem 2.2.2. However, following Hollander (1970) , we consider only equally spaced common regression constants resulting from setting c^ = c^ = c, L^ = L^ = L in (2.4.2), with n observations per line at each of the 2k x-values. Clearly then, N = 2kn, a^ = c^ (4k^-l) /12 , and H^: B^_2(N) = 6^_2(2kn) = [o)/ (2kn) ^] [2 (3^) /c (4k^-l) ^] . (2.4.18) 47 The PARES of the test procedures will be derived under two different schemes for allowing the sample size N = 2kn to tend to infinity: Ca.se 1; Let C be a positive finite constant and let c = c such that kc, ^ C as k ^ ». Consider n fixed, ^ Case_2: Let c be constant. Consider k fixed, n -> «> , resulting in a PARE in terms of k. Then let k ■> <» in this expression. Case 1 essentially allows the number of distinct regression constants (2k) to tend to infinity over a fixed region of experimentation. Case 2 considers the number of distinct regression constants fixed while the number of replicates (n) tends to infinity. The efficiency expressions derived under case 1 and case 2 will be seen to be identical. In view of (2.4.18), the rate at which the sequence of alternatives converges to the null is inversely proportional to the square root of the sample size, N = 2kn, for both case 1 and case 2 . Before proceeding to derive the various PARE's, we state a theorem due to Noether (1955) as presented by Randies and Wolfe (1979, p. 147). Theorem 2.4.1 (Noether ' s Theor em) . Let {S ,.,} and ~ — • n (x) {T^,(i)} be two sequences of tests, with associated sequences of numbers { Pg (n (i) ) ^^^ > ' ^ '^T (n' (i) ) ^ ' ^ ^ ' 48 2 2 ^^S(n(i)) ^^^ ^' ^^^ ^"T(n(i)) ^®^ ^' ^^^ satisfying the following Assumptions A1-A6: M. 'n(i) - ^S(n(i))(Qi) .^^ ^n-(i) " ^T(nWi))<^) ''S(n(i))(^^ ^T{n'(i))^'i^ have the same continuous limiting (i ^ ») distribution with cdf H(.) and interval support when 6. is the true value of •it: A2. Same assumption as in Al but with 0. replaced by 6 throughout. 1^00 (J ... (ft \ -i -i.m ^ /n \,— 1 ^S(n(i))(Qo) i-^- ^T{n'(i))<%^ A4. ^ de f'^S(n)(^^] = ^s(n) <^) ^"^ fe f^T(n')^^^^ = ^T(n') ^^^ are assumed to exist and be continuous in some closed interval about 9 = e^, with V^^^^ie^) and y'(^.)(eQ) both nonzero. A5. li^ - S(n(i))'-i^ ^ .^ ^ T(n'(i))'"i^ S(n{i)) ^^0' "^"' ^ T{n'(i)) ^'O' ^6 lim ^'s(n) ^Qq) ^'^' n-°° 2 T = K and K(„)'^o"' 49 lim ^'T(n') ^^o^ ■ , , 2 J- ~ ^T ' where K^ and K^ are positive constants, called the efficacies of the tests based on S^ and T^, respectively. Then the PARE of S relative to T is PARE (S,T) .= -\ . (2.4.19) ^T Proof ; See Randies and Wolfe (1979, p. 147). Note that assuming the equally spaced regression constants described just prior to (2.4.18) it follows that conditions I and II hold under either case 1 or case 2. Hence we need only explicitly assume conditions III and IV to apply Theorem 2.2.2 which establishes the asympototic distribution of the Sievers-Scholz statistic under H . We N now present the PAREs, PARE (Sievers-Scholz, Hollander ) Result 2.4.2; Assume the sequence of alternatives {H } (2.4.18) and the equally spaced regression constants described just prior to (2.4.18). Also assume conditions III, IV, and I(h)<". Then the PARE of the Sievers-Scholz statistic T(0) with respect to Hollander's W under case 1 or case 2 is 50 2 PARE(Sievers-Schol2, Hollander) = ^[ '^ ^ ^^^ ] . (2.4.20) ■^ 21^ (h) Verification of Result 2.4.2 ; We apply Noether's theorem. Assumptions A1-A6 must hold for T(0) and W. For 2 T(0)/a , Theorem 2.2.2 establishes Al and A2 with the standard normal limiting distribution and standardizing constants (suppressing the various subscripts and taking e=6^_2) y (9) = 4knl (g) 6 , and a^{8) = 8kn/[c^ (4k^-l)] . Then A3 , A4 , and A5 follow immediately from the form of these standardizing constants. Assumption A6 holds and efficacy (T(0)) = lim y ' (0) / [ (2kn) '^o (0) ] Itlig) case 1, (2.4.21) c(4k^-l)^I(g) case 2. We note that W is the Wilcoxon signed rank statistic computed using nk independent random variables with the distribution of ^l"^2"^3^^4 '1-2 ^ k^ 51 As in Randies and Wolfe. (1979, p. 165-166), the assxunptions A1-A6 can be validated using the equivalent statistic nk W/ (^ ■") , in which case y(6) = (2/(nk-l)) [1-H (kc (-6) ) ] + 1 - /H(kc(-t-2e))dH(kct) , o^(e) = l/(3kn) , the limiting distribution in Al and A2 is the standard normal , and efficacy (W) = lim y ' (0) / [ {2kn) ^a (0) ] N->-<» 6 ^CI (h) case 1, J, 6 ^kcl (h) case 2 . (2.4.22) Then (2.4.20) follows immediately from (2.4.19) of Noether's theorem. The first four rows of Table 1 show the value of the PARE (2.4.20) when the error distribution F is uniform, normal, double exponential, and Cauchy. The distributions are listed in order of the increasing heaviness of their tails, using the measure F-^.95) - F-^.50) ^ ^.^^_23) F ^ (.75) - F~^ (.50) 52 Table 1. PARE (Sievers-Scholz , Hollander) and Heaviness of Tails for Selected Error Distributions. Distribution Uniform Normal Double Exponential Cauchy PARE 1.29 1.33 1.48 2.67 Heaviness of Tails 1.80 2.44 3.32 6.31 CN(e,a^,5,.9) CN(e,a^,5,.8) CN(e,a^,5,.7) CN(e,a^,5,.6) CN(e,a^,5,.5) CN{e,a^,10,.9) CN(e,a^,50,.9) CN{e,a^,100,.9) CN(e,a^,500,.9) CN(e,c^,1000,.9) 1.71 1.94 1.98 1.86 1.70 2.13 2.83 2.96 3.07 3.08 2.82 4.10 5.16 5.34 5.02 3.07 3.68 3.92 4.51 4.78 1) Heaviness of Tails = F"^(.95) - f"^(.50) f"^(.75) - f"^(.50) 2) CN(e,a ,ic,t) represents the mixture of two independent normals. The first, chosen with probability x, has mean 2 6 and variance a'. The second, chosen with probability 1-T, has mean 2 2 and variance k a 53 defined by Crow and Siddiqui (1967), where F~-'-[F(t)] = t. It is clear that for these four distributions, the PARE of the Sievers-Scholz statistic with respect to Hollander's statistic increases with increasing heaviness of tails. To see whether this behavior persists for other distributions, we examined the PARE (2.4.20) for a variety of contaminated normal distributions defined as follows. Let Z = X with probability t and Z = Y with probability 1-T, where X is N(e,a'") and Y is independently N(e,K^a^). Then we say Z is distributed as a scale contaminated com- pound normal which we designate by CN (6 , a^ , k , t) . VJhen F is 2 the CN(e,a ,k,t) distribution, straightforward computations yield the following formulas for 1(g) and 1(h): ,4, 4-i ,4> 4-1, T ,1 (.)t (1-t) Kg) = -T^— J {-^^- Y— r^ ' (2.4.24) TT^a ,tr, [2(i(K^-l)+4)] ' , g (J)tS-Ni-t)^ 1(h) = -^^^— ; {-^ J-} . (2.4.25) ^'a /I [2(i(K^-l)+8)] ' In addition to the four common error distributions. Table 1 shows the value of the PARE (2.4.20) for several contaminated normal error distributions along with the heaviness of the tails of these distributions. The formiulas (2.4.24) and (2.4.25) were used to compute these PAREs while iterative techniques provided the heaviness of tails of the 54 various contaminated normal error distributions. Figure 1 shows a plot of the PARE (2.4.20) against the heaviness of tails for all the error distributions in Table 1. The main purpose of Table 1 and Figure 1 is to show the PARE (2.4.20) of the Sievers-Scholz statistic T(0) to Hollander's W is greater than one over a wide range of underlying distri- butions. Secondarily, we notice in Figure 1 that although no exact relationship exists, distributions with heavier tails tend to show higher PAREs. Under case 2, using the efficacies in (2.4.21) and (2.4.22), it follows that the PARE of the Sievers-Scholz statistic to Hollander's statistic before allowing k to tend to infinity is 4k^-l r I^(q) i 2 '■ 2 ^ ' 3k 21 (h) Thus we see that the PARE is increasing in k, with a minimum 2 2 . . value of I (g) /2I (h) for k=l . This is likely due to the fact that Hollander's procedure uses a decreasing proportion of the available slope estimates as k ->• «>. We have chosen to present our results after allowing k ->- °° for ease in interpretation and to avoid the need for a separate discussion of case 1 and case 2. PARE (Sievers-Scholz , Rao-Gore) Result 2.4.3 : Assume the sequence of alternatives {H } (2.4.18) and the equally spaced regression constants Figure 1. Plot of PARE (Sievers-Scholz , Hollander) versus heaviness of tails for selected error distributions. 1) Heaviness of Tails = F -^(.95) - f"^(.50) F~^(.75) - f"^(.50) 2) U = Uniform N = Normal E = Double Exponential C = Cauchy 2 3) CN(e,a ,k,t) represents the mixture of 2 independent normals. The first, chosen with probability t, has mean e and variance a . The second, chosen with probability 1-T, has mean 6 and variance k^o^. T^ = CN(9,a^,5,.9) T^ = CN(0,a^,5,.8) T3 = CN(e,a^,5,.7) T^ = CN(e,a^,5,.6) T^ = CN(e,a^,5,.5) K^ = CN(e,a ,10,.9) ^2 = CN(e,a^,50,.9) K3 = CN(e,a^,100,.9) K^ = CN(e,a^,500,.9) <5 = CN(6,a^,1000,.9) PARE I J. 5 +. 56 2.5 ■»• Kit — K5 T4 1.5 + 0.5 + TS Heaviness of TpIIs -'^F'»wi»w^w^gWB»»»W'^Ba8waic«iMBaw »MML i ai « ». i T ii|ii tu-uj- i M i[i^ j» l>i iaij i j ^'MW[;^B^ 57 described just prior to (2.4.18). Also assume conditions III and IV. Then the PARE of the Sievers-Scholz statistic T(0) with respect to the Rao-Gore statistic U under case 1 or case 2 is PARE (Sievers-Scholz, Rao-Gore) = 4/3. (2.4.26) Verification of Result 2.4.3 ; As in Randies and Wolfe (1979, p. 170-171), the assumptions A1-A6 of Noether's theorem can be validated for the statistic U + (nk (nk+1) /2) , equivalent to U, in which case y(e) = (nk)^/G(kc(t+e))dG(kct) + (nk (nk+1) /2) , a^(e) = n^k^/6, the limiting distribution in Al and A2 is the standard normal , and efficacy (U) = lim n ' (0) / [ (2kn) ^a (0) ] 3 '01(g) case 1, (2.4.27) 3 ''kcl (g) case 2. We showed A1-A6 hold for T(0) in the previous verification and gave the efficacies of T(0) in (2.4.21). Thus (2.4.26) follows imiD.ediately from (2.4.19) of Noether's theorem. ® 58 PARE(Sievers-Scholz, Classical Least Squares) Result 2.4.4 : Consider the classical least squares theory t-test, as specified by Hollander (1970) . Assume the sequence of alternatives {E^} (2.4.18) and the equally spaced regression constants described just prior to (2.4.18). Also assume conditions III, IV, and o 2 a = Var(E^)<«>. Then the PARE of the Sievers-Scholz statistic T(0) with respect to the classical t-test statistic under case 1 or case 2 is PARE (Sievers-Scholz, classical) = 24a^[I^(g)]. (2.4.28) Verificat ion of Result 2.4.4 ; Let B^ and ^2 denote the least squares estimates of 6^ and B^' respectively, and let 2 s be the residual mean square error (see (3.1) of Hollander (1970, p. 389)). Then the form of the t-test statistic used to test Hq: Bj^_2 = under the specified equally spaced regression constants is e - 6 ^ - 1 2 t - 2 • (2.4.29) r 12s ,h 2 2-' nkc (4k^-l) Using the fact that s is a consistent estimate of a^ under B^^ and a discussion similar to that in Randies and Wolfe (1979, p. 164-165), it follows that assumptions A1-A6 of Noether's theorem hold for the statistic t with iJiii II WH»*Wiui I " wii ■ ■ "I, j-ionwif ij .11 iiiH» 59 y{e) = ^- ' ^"'^ (2.4.29) r 12a^ .h ^ 2 ? ^ nkc (4k -1) a^(e) - 1. The limiting distribution in Al and A2 is the standard normal and efficacy (t) = lim y ' (0) / [ (2kn) 'a (0) ] [ — 5-] case 1, 6a , c^(4k^-l) ,^ [ ^ 5—^] case 2. 24a^ (2.4.30) A1-A6 hold for T(0) as shown in the verification of Result 2.4.2 with the efficacies given by (2.4.21). Thus (2.4.28) follows immediately from (2.4.19) of Noether's theorem. The efficiency (2.4.28) is identical to the familiar PARE of the two-sample Wilcoxon-Mann-Whitney test with respect to the two-sam.ple normal theory t-test when the cdf of the underlying error distribution is G. Hence, the PARE (Sievers-Scholz, classical) > 0.864 for all G (see Hollander (1970) for a proof that in this case the inequal- ity is strict). Also, this PARE equals 0.955 when G is normal and is greater than one for many non-normal G. 60 PARE (Sievers-Scholz.Sen) Result 2.4.5: Assume the sequence of alternatives {H,^} (2.4.18) and the equally spaced regression constants described just prior to (2.4.18). Also assum.e the following: 1) Conditions III and IV. 2) I* (F) < ~ as in (2.4.11), where F is the under- lying error distribution. 3) The score function ({) (u) is an absolutely continuous and nondecreasin.g function of u, < u < 1, that is square integrable over (0,1). Let pCy,!})) be defined as in (2.4.13). Then the PARE of the Sievers-Scholz statistic T(0) with respect to Sen's (1969) statistic L under case 1 or 2 is PARE (Sievers-Scholz, Sen) = J"^ -*• ^^^ ' (2.4.31) p^ (¥,<!)) I* (F) Verification of Result 2.4.5 : Sen's statistic L has an asymptotic (N->-<») chi-squared distribution under H (Sen, 1969, p. 1676). Using results _* from Sen's work we define a statistic, V , that has an asymptotic normal distribution under H and whose square (multiplied by a constant) is asymptotically equivalent to L _* under H . Since the square of V (multiplied by a constant) has the same asymptotic distribution under H as L, we use _* V in applying Noether's theorem to derive efficacy ?^s^sa>B9!gi^~ap;> i i -i ji gi pt* ' »n, <, !■ hjiuhii 61 expressions for Sen's test and the PARE (Sievers-Scholz , Sen) . Let B^ denote the Hodges-Lehmann estimate of 3, based on V^ (2.4,10), i=l,2, and let 6* = 1/2(6* + 6*)* From equations (3.10) and (3.20) in Sen (1969, p. 1673, 1675) it follows that a = N'a (6 -3*) = 0^(1) and ^ J. p as N^" under H^, i=l,2. Using these definitions of a and b and the notation (2.4.14) we apply Leimna 3 . 2 in Sen (1969, p. 1674) , which is given in Chapter Three of this work as Lemma 3.4.3, yielding V.(Y.-B X) - V.(Y.-6.x) = N'a^(6*-3*)p(¥,(|,) [I*(F)]^ + o (1) as N-^- under H^. Equation (3.22) in Sen (1969, p. 1675) states |Vi(Y.-e*x)| = o (1) •^'"'''Wl f .l ii . ' l"_ l «.>--'_P_» i|«l . l lv « LM i«» 62 and so, applying this to the previous result. ^i " Vi(Ii-^*2S) = N'a^(S*-6*)p(Y,<(,) [l*(F)]'^ + o (1) as N->co under H^. Applying equation (3.19) in Sen (1969, p. 1675) , N^a^{6*-3*) = o (1) , we have ^i " N'a^(B%B*)p (T,^) [i{F)]^ + o (1) as N^-<» under H^. It follows that 63 2 L = p^(Y,<f)I*(F)NaJ E (e*-3*)^ + o^(l) ^ i=l ^ P = p^(^,(j))I*(F)Na^%(3*-32)^ "^ °p^^^ as N->» under H„. We define the statistic N -* ■ H. * * V = (N/2) cr^(3-^-62) (2.4.32) and note that the asymptotic distribution of 2 * -* 2 p ('!',(()) I (F) (V ) under H is the same as that of L. From Lemma 3.4 in Sen (1969, p. 1676), p(^,<i>) [I*(F)]^[V*-(N/2)'^a^(6^-S2)] t N(0,1) as N-»-« under H . Thus assumptions A1-A6 of Noether's _* theorem hold for the statistic V with y(0) = [c(4k^-l) V(2(3'))] G, and o^ie) = [p^(^,<^)i^ (F)] ^ 64 The limiting distribution in Al and A2 is the standard normal and efficacy (V ) = lim y ' (0) / [ (2kn) ^o (0) ] h, ,.% [Cp(T,.^) (I (F))^]/6^ case 1, (2.4.33) .[c {4k^-l) ^p (¥,<(,) (I* (F) ) '^j / [2 (6^) ] The assumptions A1-A6 hold for T{0) as shown in the verification of Result 2.4.2 with the efficacies given by (2.4.21). Hence (2.4.31) follows immediately from (2.4.19) of Noether's theorem. To evaluate the expression (2.4.31), we first note the following, which results from the definition of A^ in (2.4.9) and q(1 ,^) in (2.4.13) : P^('i',<).)I*(F) = [ / ¥(u)(j,(u)du]2/A^ ^ 2 1 = [ / >l'(u)(|)(u)du]"^/[ / [ Mu) -<()*] ^du] . (2.4.34) Suppose we assume Mu) = u, < u < 1. Scores resulting from this choice of score function are called Wilcoxon 65 scores. Direct computations show that in this case 1 2 / [(j)(u) - ^*]^du = 1/12. (2.4.35) Using a derivation similar to one in Randies and Wolfe (1979, p. 308) we show that when ^ (u) = u, the numerator of the right-hand side of (2.4.34) equals I^(f): / Y(u)(|) (u)du = - / u[f' (F "-(u) )/f (F ^(u))]du. (2.4.36) Let t = F (u) , resulting in 1 Ul) / 4'(u)(i. (u)du = - / F(t)f'(t)dt, (2.4.37) UO) where F(?(p)) = p. Now let u = F(t), dv = f'(t)dt and apply integration by parts to (2.4.37): ^ Ml) ^^^) 2 / ^(u)((.(u)du = - {[F(t)f (t)]^ ; - / f^(t)dt} ^^^> UO) = [F(UO))f (5(0))] - [F(C(l))f (UD)] + / f^(t)dt. 5(0) (2.4.38) If we assiime the support of F is [a,b] where a < b and f(x) ^-Oasxfborx + a, then we take 5(0) = a, 5(1) = b in (2.4,38) and 66 1 b / Y (u) (}) (u) du = / f^ (t)dt = 1(f) . (2.4.39) a As Randies and Wolfe (1979, p. 313) state, this same form (2.4.39) can be obtained under more general assumptions. Hence substituting (2.4.35) and (2.4.39) into (2.4.34) it follows from (2.4.31) that under ^ (u) = u, H , and assumed regularity conditions, 2 PARE(Sievers-Scholz, Sen) = ^^ ^^^ . (2.4.40) I (f) Table 2 gives values of the PARE (2.4.40) for four common error distributions. We see that when compared to the Sen statistic using Wilcoxon scores, the Sievers-Scholz statistic achieves a PARE close to (or equalling) one for error distributions having light to moderately heavy tails (uniform, normal, double exponential) . However the Sievers-Scholz statistic has poor PARE under the Cauchy distribution which has very heavy tails. That the asymptotic performance of Sen's test is better than other tests is not surprising since Sen's test is the rank test that maximizes the efficiency relative to the likelihood ratio test. However, Sen's test not only requires iterative calculations, but is distribution-free only asymptotically. More discussion of the relative merits of Sen's test and the Sievers-Scholz testd will be given in Section 2.6. 67 Table 2. PARE (Sievers-Scholz , Sen) for Selected Error Distributions Assuming (j) (u) = u. Distribution Uniform Normal Double Exponential Cauchy PARE (Sievers-Scholz, Gen) 0.89 1.00 0.78 0.50 Summary of PARE Results In conclusion, PAREs derived under equally spaced regression constants favor the Sievers-Scholz approach over the other two exact, nonparametric competitors due to Hollander and Rao and Gore. Specifically, the PARE of the Sievers-Scholz statistic with respect to Hollander's statistic is greater than one over a wide range of underlying error distributions. Indeed this efficiency frequently exceeds two. Even more interesting is the fact that the PARE of the Sievers-Scholz statistic with respect to the Rao-Gore statistic is 4/3 for all error distributions (subject to certain regularity conditions required to derive the PARE) . The Sievers-Scholz statistic achieves the familiar PARE (2.4.28) when compared with the classical least squares theory t-test. Although the PARE of the Sievers-Scholz statistic with respect to Sen's statistic is 68 less favorable under the heavily tailed Cauchy distribution, we noted that Sen's test is distribution-free only asymptotically. V7e discuss these two methods further in Section 2.6. Note that under equal spacing, the PARE of the Sievers- Scholz statistic to the Theil-Sen statistic (2.1.4) dis- cussed in Section 2.1 is one (Sievers, 1978). Thus all of the previous ARE comparisons also apply when using the (zero-one weighted) Theil-Sen statistic in place of the Sievers-Scholz statistic. The advantage of the Sievers- Scholz approach over that of Theil and Sen will appear in the Monte Carlo comparisons under unequal spacing of the regression constants discussed in Section 2.6. 2.5 Small Sample Inference Since the test of parallelism and the confidence intervals for g^^^ presented in Section 2.3 depend on asymptotic theory, they are generally only applicable with moderately large samples. In this section we discuss exact, distribution-free tests of H^ : g^^^ = and corresponding exact confidence intervals for the slope difference, B These tests continue our basic approach of applying the method of Sievers and Scholz to the two line setting, again assuming comm.on regression constants. Specifically, we discuss two related techniques. The first utilizes the exact distribution of the Sievers-Scholz statistic TiO) with the optimum weights, a = x , under 69 the null hypothesis H^ : 3-,_2 = 0. This null distribution depends on the chosen regression constants, and hence must be recalculated for each design. The second technique is a straightforward application of the Theil-Sen approach to the two line setting, as discussed in Section 2.1. Since the required null distribution is essentially that of Kendall's o tau, tabled critical values are readily available for small sample sizes. Thus the second technique has favor under sample sizes sm.all enough to discourage the use of asymptotic results, yet too large to allow the computation of the null distribution of T(0) required by the first technique. Let us now discuss in detail the two small-sample techniques we have proposed. First, recall the basic linear model (2.1.1) we have assumed, the resulting distribution of the differences Z. given by (2.1.3), and the representation of T(0) in terms of the ranks of the Z's given by setting b = in (2.2.1) : N T(0) = (2/N) Z [Rank(Z.)x ] - (N+l)x. (2.5.1) j = l ^ ^ Suppose that for line i (i=l,2) the underlying errors, E. ., j=l,...,N, are independent and identically distributed (i.i.d.). The distribution of the line 1 and line 2 errors need not be the same, but assume these two distributions are continuous. Let ^ denote the set of N! permutations of 70 (1,2,...,N). Then under H^ : 3-|^_2 = 0, the Z's are i.i.d. and hence the vector of ranks of the Z's is uniformly distributed over ^^ (Randies and Vvolfe, 1979, p. 37). Consequently, in view of the representation (2.5.1), the null (B^_2 = 0) distribution of T(0) is uniformly distributed over the N! values of T(0) obtained by permuting the ranks of the Z's among themselves. For example, if for r = (r^,. . . ,r^) e|^, N t„ = [ (2/N) E (r.x.)] - (N+l)x , - j=l ^ ^ then the distribution of T(0) under H. : g = is given by Pq {T(0)=t^} = 1/N!, rti^, (2.5.2) where, as before, ^^i ' } denotes the probability calculated assiiming B^^^ = te Having tabulated this null distribution, a test of H- is conducted as follows. If t^ is a constant determined from (2.5.2) such that Pq {T(0)>_t^} = a, then the test of H^ : &^_^ = against Hq: B^_2 > which rejects H„ if T(0) 1 t^ (2.5.3) is an exact level a test. Rather than simply state whether a test has accepted or rejected a null hypothesis at a i MU i u 'ww LHJ ^ - ' j ^ tig ii j iw ^i^ tt Mw u ' L iii ^T, v ftai ri ^Sf* 71 particular level, one might wish to report the attained significance level, the lowest significance level at which the null hypothesis can be rejected with the observed data, If we observe T(0) = t(0), then the exact test (2.5.3) has an attained significance level of p = [ (number of r in P) /N! ] , where P = {re^,: t(0)£t^}. Two-tailed tests can be obtained by the usual modifications. Lst t^ -2 be a constant defined analogously to t : ^0 ^-^a/2^^(0)^^a/2> = 1 " - Applying the argximent and notations used to derive (2.3.3), it is easily seen that [j;^(%(l-(Nt^/2/^--)))' jZ^(^a+i^t^^^/x..)))) (2.5.4) is an exact 100 (l-a) percent confidence interval for 6. In terms of the slopes S^^ = (Z -Z ) /x , 1 j< r < s <_ N, we can write this interval as [S""^, S^) , where Sg = min{S^g: J (S^^) >% (1- (Nt^^2/^- • ) ) ' l<r<s<N}, and MB'.m^.r^Tl'^igi l 72 Sq = min{S^^: J (S^^) >h {1+ (^t^^^/x. .) ) , l.<r<s<N} . (2.5.5) Computation of a null distribution based on permuta- tions of the observations (or ranks) such as the one in (2.5.2) has become possible under small sample sizes with the speed of today's computers. At the University of Florida we found that the necessary calculations were feasible when there were N=8 or fewer regression constants per line. In this case the entire null distribution of T(0) could be tabulated at a cost of about $2.00 to the user. We were accessing a system operating an IBM 3081 with MVS/XA and an IBM 3033 with OS MVS/SP JES2 Release 3. With more than 8 regression constants per line, the cost of tabulating the null distribution becomes prohibitive (at least $30.00 when N=10, for example). However, in dealing v/ith similar problems, Pagano and Tritchler (1983) and Mehta and Patel (1983) give algorithms that greatly reduce the amount of computation involved. Although their results do not apply directly to this problem, we anticipate the exact test (2.5.3) and computation of the confidence interval (2.5.4) will soon become feasible for larger sample sizes due to the development of similar efficient algorithms and the steadily increasing speed of computer hardware. At the present time we could resort to estimation of the null distribution (2.5.2) of T(0) based on a random sample of permutations of the observed rank vector. Boyett ■ "" i^u f^ gJC .1 1 ii a^ i qm BBj rL P'!WqiMWi^-*?j g, 'g"' . » ajt'j » i ■■ wn nn mngi w em w * t atm m nuiim 73 and Shuster (1977) and Ireson (1983) report good approxima- tions resulting from the use of such a sampling of permu- tations to approximate a permutation distribution. However, there is a possibility that approximate procedures such as these suffer a loss of power due to the restricted sample space (Dwass, 1957) . Also, there is the problem of specifi- cation of the number of permutations that must be sampled to achieve adequate approximations of the null distribution. Clearly, more study is needed before this technique can be recommended without reservation. Another method of overcoming the computational problem associated with the exact test under larger sample sizes is to replace the Sievers-Scholz statistic by the Theil-Sen statistic based on the differences Z . , as described in Section 2.1, Recall that this consists of using the weights ^rs " sgn(x^^) in the expression for T (b) given by (2.1.2). The null distribution is essentially that of Kendall's tau, which has been tabulated for many values of N. These tabled critical values and a precise specification of the Theil-Sen test and confidence interval are given in Hollander and Wolfe (1973) . This second approach is appropriate when an exact, distribution-free technique is desired but the number of regression constants per line is too large (N>8, at our facility) to allow complete enumeration of the null distri- bution (2.5.1) of T(0). If the regression constants are highly unequally spaced, the method based on Sievers-Scholz procedure probably has greater power, but the results of a 74 simulation study discussed in the next section indicate that in many situations the difference in power of the two techniques is slight. One detail that has not been mentioned in this section is how to deal appropriately with ties in the data. If ties occur among the Z's when applying the first technique, simply use all permutations of the vector of midranks in the computation of the null distribution of T(0). The null distribution computed is the conditional null distribution of T(0), given the observed midranks, and the exact, dis- tribution-free properties of the test and interval are retained. We assume the Theil-Sen approach is only being applied when the sample size prohibits computation of the exact null distribution of T(0). In this case, modifica- tions of the Theil-Sen approach based on Kendall's tau in the presence of ties are referenced by Hollander and Wolfe (1973, p. 192), but they do not retain the exact nature of the test and confidence interval. As discussed in Section 2.4, the Hollander and Rao-Gore methods also provide exact, distribution-free tests of HqJ 3^_2 = and confidence intervals for the slope differ- ence, 3-j^_2. The null distributions required to use their methods have been tabulated and are readily available for several sample sizes. Hence these methods are competitors to the exact, small sample techniques proposed in this section. Monte Carlo comparisons of the power of these tests are given in the next section. 75 2.6 Monte Carlo Results To compare the powers of the test statistics discussed in the previous sections, we conducted a Monte Carlo study. The study concentrated on unequally spaced regression constants for the following reasons. The asymptotic rela- tive efficiency results in Section 2.3 indicate superiority of the Sievers-Scholz procedure in a wide variety of cases when the regression constants are equally spaced. Since the structure of the Sievers-Scholz statistic T (b) (2.1.5) utilizes information about the spacing of the regression constants, one would expect that its relative performance improves when unequal spacing is used. Hence, we were particularly interested in comparisons of the test statis- tics under unequal spacing of the regression constants. We begin our discussion with a description of the sample sizes, regression constants, error distributions, and parameter values used in our simulation study. Choice of Regression Constants Recall from Section 2.4 that Hollander's technique requires a scheme for pairing the regression constants on each line to form slope estimates. Although Hollander clearly specifies the pairing scheme (2.4.3) under equal spacing (2.4.2) of the regression constants, there exists some ambiguity in the choice of such a scheme when the regression constants are unequally spaced. To avoid this 76 ambiguity, we use what we call mirrored spacing, which we now describe. Consider a smooth nondecreasing function t(-) defined over the interval [0,1/2]. Suppose t{.) maps this interval onto itself with t(0) =0 and tCl/2) = 1/2. Then we define 2k regression constants, x^, x^, ..., x^j^, over the interval [0,1] by the relations "m = ^(k?l) ' ^m+k = ^^m + 1/2 ' for m = 1, ..., k. Multiplication by a scale factor can be used to make these constants fall over a more natural range. We use the term mirrored spacing since, by definition, 'm+k ~ ■^m "^ "'■/^ for ^ = 1/ , k, and so the arrangement of the set of x's {x^^^^, ..., x^^^} over [%,!] is identical to the arrangement of the set {x^, ..., x^^} over [0,^]. when constructing Hollander's statistic we pair responses at x m with those at x^^^, thus conforming to Hollander's recom- mendation to choose a pairing scheme approximating that used with equal spacing. Mirrored spacing also allows the use of the Rao-Gore statistic, which is not generally applicable with unequally spaced regression constants. We experimented with t(.) of the form t (u) = au^, where a is chosen to satisfy t(l/2) = 1/2 and i is a positive integer. Note that i = 1 results in two groups of equally X 77 spaced constants over the intervals (0,1/2) and (1/2,1). As i increases, the regression constants tend to group closely just below 1/2 and 1. For our simulation study we selected 1-3, t(u) = (1/3) 'u . This choice of i results in regres- sion constants that depart sufficiently from equal spacing without the excessive clumping observed under larger values of i. Selection of Err or Distributions and Parameter Values To generate simulated random variates we used the Fortran subroutines of the International Mathematical and Statistical Library (IMSL) . We selected four error dis- tributions: uniform, normal, double exponential, and Cauchy. In terms of the heaviness of tails, these dis- tributions cover a broad range. They are listed above in order of increasing heaviness of tails, from the uniform distribution which has very light tails to the Cauchy distribution which has very heavy tails. The standard normal distribution (N(0,1)) was used, and scale factors of the other distributions were selected such that the proba- bility between -1 and 1 was the same for all four distribu- tions . The values of &^_^ at which the power was estimated were determined by selecting multiples of the estimated standard deviation of the difference of the least squares estimates of g^ and e^ in each case. These multiples were chosen to achieve a wide range of power. 78 Designs and Computational Details There are basically two parts to the Monte Carlo study presented here. In the first part we used moderately large samples and applied approximate tests based on asymptotic theory. The second part of the simulation used small sample sizes and exact nonparametric tests. The form of the classical t-test was the same in both parts of the Monte Carlo simulation. For the first part of the Monte Carlo study, dealing with moderately large samples, we chose two designs which we call design A and design B. In presenting the sample sizes used in each design, we give the number of regression ■constants per line. Since we are assuming two lines, the total sample size is twice of what we give below. Design A consists of 3 replicates at each of 20 distinct regression constants, resulting in 60 observations per line. The 20 distinct regression constants were selected by using mirrored spacing as described and multiplying by a scale factor of 20. The resulting regression constants are listed in Table 3. Design B uses 30 distinct regression constants, with one observation per line taken at each of these constants. The constants for design B were selected by generating 30 random numbers between zero and one, and multiplying them by a scale factor of 30. We wanted one design chosen to allow power comparisons of the tests under regression constants whose spacing followed no structured pattern, and 79 Table 3. Regression Constants Used for Designs A and B, Design A Design B X. X, X, X- ^10 X, X, X, X,. X, X. Xr X, X X X '10 11 12 'l3 14 15 = 0.0075 = 0.0601 =. 0.2029 = 0.4808 = 0.9391 = 1.6228 = 2.5770 = 3.8467 = 5.4771 = 7.5131 = 0.65 = 1.16 = 5.46 = 6.09 = 8.34 = 8.52 = 9.50 = 10.83 - 13.61 = 13.84 = 13.93 = 15.20 = 15.42 = 15.58 = 16.51 X X X X X X 11 '12 13 14 15 16 17 18 19 X 20 X X X X X X X 16 17 18 19 20 21 22 23 X X X '24 '25 26 27 28 '29 '30 = 10.0075 = 10.0601 = 10.2029 = 10.4808 = 10.9391 = 11.6228 = 12.5770 = 13.8467 = 15.4771 = 17.5131 = 17.59 = 18.81 = 18.83 = 19.53 = 19.99 = 20.13 = 20.78 = 21.01 = 22.76 = 23.78 = 25.86 = 26.37 = 26.59 = 28.06 = 28.92 1) For Design A, 3 responses per line were observed at each of the 20 regression constants. 2) For Design B, 1 response per line was observed at each of the 30 regression constants. 80 hence this accounts for our departure from mirrored spacing in this case. The regression constants for design B are listed in Table 3. To compute Hollander's statistic under design B, we used the usual grouping scheme, pairing the response at x^ with that at ^^+^5' ^^^^^ m = , 1, ..., 15. Note that given the arbitrary spacing of the regression constants, the Rao-Gore statistic is not applicable for design B. All tests applied in the simulations under design A and B used a nominal level of a = .05, and in each case the null hypothesis H^ : ^-^-2 ~ ° ^^^ being tested against the one- sided alternative H^ : ^^_^ > 0. The tests used were the Sievers-Scholz test described in Section 2.3, the Theil-Sen test (based on T* (b) in (2.1.2) with a = sgn (x -x ) ) , the Hollander test based on the statistic W defined in Section 2.4, the Rao-Gore test (design A) based on U defined in Section 2.4, and the t-test based on the difference of the least square estimates of 6, and g-- Each of these tests were employed at an approximate a = .05 level utilizing their respective asymptotic distributions. For comparisons of the various procedures under small samples we selected three designs, which we call design C, design D, and design E. All three designs consist of one response per line at each of the regression constants. Sample sizes per line of 6 , 8, and 12 were used in designs C, D, and E, respectively. Mirrored spacing was 81 used for all three designs. The design points were multi- plied by scale factors of 10, 10, and 15 for designs C, D, and E, respectively. The resulting regression constants for these three designs are given in Table 4 . When using designs C and D, the exact tests of ^0" ^1-2 ^ ° against H^ : &^_^ > Q associated with each of the four nonparametric procedures were used. The exact Sievers-Scholz and Theil-Sen tests as discussed in Sec- tion 2.5 were applied. Thus the exact distribution of the Sievers-Scholz statistic T{0) was first computed for designs C and D to determine appropriate critical values. An example of a portion of this distribution for design C is given in Table 5. Exact versions of the Hollander and Rao-Gore tests rely on the exact null distributions of the Wilcoxon signed rank and Wilcoxon-Mann-Whitney statistics, respectively, as tabulated and discussed in Hollander and Wolfe (1973) . Randomization was used to bring these exact procedures to the same level. The natural a-levels of the tests were compared to select a level for each design at which the amount of randomization needed was minimal. A nominal level of a = .125 was used for design C while a = .057 was selected for design D. With 12 observations per line, design E does not allow the feasibility of the exact Sievers-Scholz test. Instead, we replaced this with an approximate procedure, randomly selecting 10,000 permutations of the rank vector and com- puting the proportion of these perm.utations resulting in a "-iM-^ti^*;, ■ |HMiH|ni(^i9i« 82 Table 4. Regression Constants Used for Designs C, D, and E. Design C ^i = 0.078 x^ = 5.078 x^ = 5.625 Xg = 7.109 Design D x^ = 0.040 x^ = 5.040 Xg = 5.320 x^ = 6.08 Xg = 7.560 Design E x^ = 0.022 x^ = 7.522 Xg = 7.675 Xg = 8.090 x^Q = 8.899 x^^ = 10.233 x^2 = 12.223 ^1 = 0.078 ^? = 0.625 ^3 = 2.109 ^] = 0.040 ^2 = 0.320 ^3 = 1.080 ^4 = 2.560 ^1 = .022 ^7 = .175 ^3 = .590 ^4 = 1 .399 ^5 = 2 .733 ^6 = 4 .723 For Design C, D, and E, one response per line was observed at each design point. ^*=«!*«¥Va3»¥^-^-^i|||)l . 83 * Table 5 . Upper Portion of the Exact Null Distribution of T(0) when using Design C. t(0) Pq {T{0) > t(0)} 6.64062 6.69269 6.82289 6.82292 6.87499 7.00519 7.13542 7.18749 7.31769 7.36979 7.49999 7.68229 7.81249 7.86459 7.99479 8.17709 0.0583 0.0542 0.0528 0.0514 0.0486 0.0472 0.0444 0.0431 0.0389 0.0347 0.0333 0.0236 0.0194 0.0167 0.0139 0.0111 -^i-t*ut,rwi»(i«w«<**»rif--»"!**'*^'F— - 84 value of T(0) larger than the one calculated from the observed data. Rejection of H^ : Q^_^ = o in favor of ^1' ^1-2 ^ ° occurred when this proportion was less than or equal to the nominal level, which was set at a = .047. Recall that this type of approximate procedure was discussed in Section 2.5. The implementation of this approximate procedure was facilitated by an algorithm due to Knuth (1973) , which presents a one-to-one association between the integers 1,...,N! and the N! permutations of (1,...,N). Because of the cost of this approximate Sievers-Scholz procedure, the number of simulations for design E was set at 500, and only two error distributions were selected (normal and Cauchy) . Discussion of Results Empirical levels and powers of the Sievers-Scholz, Theil-Sen, Hollander, Rao-Gore, and classical tests under design A are presented in Tables 6, 7, 8, and 9, for the uniform, normal, double exponential, and Cauchy distribu- tions, respectively. Empirical levels vary but generally remain within two standard errors of the nominal .05 level. It is seen that in terms of their power, the Sievers-Scholz and Theil-Sen tests uniformly dominate the Rao-Gore and Hollander tests. The power of the classical t-test is highest under uniform and normal errors, but falls below the powers of Sievers-Scholz and the Theil-Sen tests for double exponential errors. The t-test has the lowest power of the 85 Table 6. Empirical Power Tiir.es 1000 Under Design A for the Uniform Distribution (a=.05). h-2 .024 .064 .096 Sievers-Scholz 058 192 675 934 Theil-Sen 059 2 - 18 14 Hollander 063 4 - 57 - 35 Rao-Gore 061 - 15 - 58 - 36 Classical 061 17 71 31 The first row and first column give the empirical power times 1000. Entries in the rem.ainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. Table 7. Empirical Power Times 1000 Under Design A for the Normal Distribution {a=.05) . Sievers-Scholz 041 217 561 861 Theil-Sen 036 5 - 21 - 15 Hollander 046 - 10 - 70 - 51 Rao-Gore 041 - 25 - 41 - 53 Classical 042 33 27 26 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicatira a lower power than Sievers-Scholz. 86 Table 8. Empirical Power Times 1000 Under Design A for the Double Exponential Distribution (a=.05) . ^1-2 .032 .064 .096 Sievers-Scholz 052 234 476 808 Theil-Sen 043 - 16 6 - 14 Hollander 057 - 46 - 69 - 116 Rao-Gore 055 - 29 - 53 - 67 Classical 049 - 16 - 32 - 22 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. Table 9. Empirical Power Times 1000 Under Design A for the Cauchy Distribution (a=.05) . ^1-2 .064 .096 .160 Sievers-Scholz 052 394 590 896 Theil-Sen 052 - 13 1 1 Hollander 053 - 140 - 245 - 275 Rao-Gore 062 - 41 - 44 - 31 Classical 043 - 289 - 451 - 667 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. 87 five tests under the heavily tailed Cauchy distribution. The Sievers-Schclz test tends to have higher power than the Theil-Sen test, but the difference is not pronounced. Comparing Hollander's test with that of Rao and Gore, we see that the power of the Rao-Gore test dominates the power of Hollander's test for the two heavily tailed distributions (double exponential and Cauchy) but they perform about equally under normal and uniform errors. A summary of all the Monte Carlo results will be given at the conclusion of this section. Tables 10, 11, 12, and 13 present empirical levels and powers of the Sievers-Scholz , Theil-Sen, Hollander, and classical tests under design B for the uniform, normal, double exponential, and Cauchy distributions, respectively. Empirical levels of the tests appear somewhat depressed, but generally fall within two standard errors of the nominal .05 level. For the uniform, normal, and double exponential distributions, the order of the tests in decreasing power is classical, Sievers-Scholz, Theil-Sen, and Hollander. The dominance of the classical approach lessens with increasing heaviness of tails of the ■ error distribution. For Cauchy errors the classical test again falls into last place, with the Sievers-Scholz and the Theil-Sen tests exhibiting highest powers. For design C, Tables 14, 15, 16, and 17 present empiri- cal levels and powers under uniform, normal, double exponential, and Cauchy distributions, respectively. Table 10 88 Empirical Power Times 1000 Under Design B for the Uniform Distribution (a=.05). ^1-2 .034 .068 .102 Sievers-Scholz Theil-Sen Hollander Classical 039 036 037 043 253 - 16 - 62 37 651 - 18 - 161 76 934 - 12 - 89 31 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. Table 11. Empirical Power Times 1000 Under Design B for the Normal Distribution (a=.05). Sievers-Scholz Theil-Sen Hollander Classical 042 038 055 046 209 20 24 25 569 - 28 - 128 48 878 18 154 26 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. ^•"•^^rv* •■•«J'f ■"** c --'I'-* ■»— ^ J- 89 Table 12. Empirical Power Times 1000 Under Design B for the Double Exponential Distribution (a=.05). Sievers-Scholz Theil-Sen Hollander Classical 047 039 045 053 201 4 - 21 18 495 - 24 - 137 2 753 - 23 - 171 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, v/ith a negative value indicating a lower power than Sievers-Scholz. Table 13. Empirical Power Times 1000 Under Design B for the Cauchy Distribution (a=.05). Sievers-Scholz 044 225 577 770 Theil-Sen 047 6 5 7 Hollander 041 - 90 - 250 - 324 Classical 043 - 133 - 377 - 455 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. •mtfimmmmmspm*'^ 90 Table 14. Empirical Power Times 1000 Under Design C for the Uniform Distribution {a=.125). Sievers-Scholz 121 424 824 977 Theil-Sen 125 - 27 - 61 - 37 Hollander 113 - 53 - 101 - 50 Rao-Gore 122 - 26 - 57 - 7 Classical 128 55 51 17 The first row and first column give the emcpirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. Table 15. Empirical Power Times 1000 Under Design C for the Normal Distribution (a=.125). ^1-2 .218 .436 .654 Sievers-Scholz 112 371 687 918 Theil-Sen 117 - 17 - 48 - 35 Hollander 110 - 52 - 88 - 91 Rao-Gore 103 - 30 - 49 - 44 Classical 117 6 58 36 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a 'lower power than Sievers-Scholz. 91 Table 16. Empirical Pcv/er Times 1000 Under Design C for the Double Exponential Distribution (a=.125). ^1-2 .218 .436 .654 Sievers-Scholz 110 354 612 854 Theil-Sen 117 - 16 - 20 47 Hollander 123 - 46 - 41 - 65 Rao-Gore 113 - 25 - 26 34 Classical 111 23 61 37 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz . Table 17. Empirical Power Times 1000 Under Design C .f or the Cauchy Distribution (a=.125). Sievers-Scholz 131 320 465 574 Theil-Sen 122 - 13 9 1 Hollander 132 - 57 - 50 - 41 Rao-Gore 125 - 42 - 25 - 11 Classical 141 - 23 2 4 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. '-n^TP'lV ^'/f-r^ t^*K-j* ^fr w^-; 92 Tables 18, 19, 20, and 21 present these same results for design D. Recall that designs C and D consisted of samples of size 6 and 8 per line, respectively, and the exact nonparametric tests were used under these designs. The relative performance of the tests using these small sample sizes is similar to that of the approximate tests under larger samples, except that the dominance of the classical t-test is more dramatic under the smaller sample sizes. Only for Cauchy errors under design D does the power of the Sievers-Scholz test clearly dominate the classical test. The Sievers-Scholz and Theil-Sen tests generally exhibited greater powers than the Hollander and Rao-Gore tests. The one exception to this occurred under uniform errors, where the observed power of the Rao-Gore test was slightly higher than that of the Theil-Sen test, although we note that in these cases the Sievers-Scholz test had highest power among these three tests. For design E, Tables 22 and 23 present empirical levels and powers for the normal and Cauchy distributions, respec- tively. When compared with the Theil-Sen test, the approximate Sievers-Scholz test had higher power under normal errors and lower power under Cauchy errors. Hence these results do not indicate a clear choice between these two methods. However, the Sievers-Scholz and Theil-Sen tests once again exhibit greater powers than the Hollander and Rao-Gore tests (except in one instance v/here the power of the Rao-Gore test marginally exceeded that of the 1 »:;»J Tl C i»i« Tll, ■I JM"'i ifrT°" 93 Table 18. Empirical Power Times 1000 Under Design D for the Uniform Distribution (c=.057). Sievers-Scholz 059 300 661 Theil-Sen 064 - 26 - 36 Hollander 056 - 68 - 160 Rao-Gore 064 - 16 - 25 Classical 057 31 81 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, V7ith a negative value indicating a lower power than Sievers-Scholz. Table 19. Empirical Power Times 1000 Under Design D for the Normal Distribution (a=.057). ^1-2 .186 .372 Sievers-Scholz Theil-Sen Hollander Rao-Gore Classical 059 056 050 054 058 223 9 - 41 - 28 34 551 - 41 - 120 - 47 94 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. =»(i««jicsc;;i— ST*Mfii 94 Table 20. Empirical Power Times 1000 Unc3er Design D for the Double Exponential Distribution (a=.057). 1-2 Sievers-Scholz Theil-Sen Hollander Rao-Gore Classical .372 .558 059 451 684 054 - 11 - 23 053 - 92 - 135 059 - 22 - 29 063 60 90 The first row and first column give the em.pirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. Table 21. Empirical Power Times 1000 Under Design D for the Cauchy Distribution {a=.057) . ^1-2 .372 .930 Sievers-Scholz Theil-Sen Hollander Rao-Gore Classical 058 057 064 057 062 342 - 17 - 71 - 28 - 65 645 17 - 112 - 63 - 44 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers- Scholz result, with a negative value indicating a lower power than Sievers-Scholz. g i Q Pi gaog i. s^j!yit*giai 95 Table 22. Empirical Power Times 1000 Under Design E for the Normal Distribution (a=.047). '1-2 100 200 Sievers-Scholz Theil-Sen Hollander Rao -Go re Classical 058 228 576 064 - 18 - 48 056 - 46 - 98 058 - 46 - 46 062 - 14 46 1) The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers-Scholz result, v;ith a negative value indicating a lower power than Sievers-Scholz. 2) All powers based on 500 simulations. 3) An approximation to the exact Sievers-Scholz technique was used by selecting a random sample of 10,000 permuta- tions of the rank vector. Table 23. Empirical Power Times 1000 Under Design E for the Cauchy Distribution (a=.047) . 1-2 .100 .200 Sievers-Scholz -Theil-Sen Hollander Rao-Gore Classical 060 146 288 062 2 36 084 - 34 - 72 046 - 12 - 18 054 - 32 - 96 1) The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding Sievers-Scholz result, with a negative value indicating a lower power than Sievers-Scholz. 2) All powers based on 500 simulations. 3) An approximation to the exact Sievers-Scholz technique was used by selecting a random sample of 10,000 permuta- tions of the rank vector. 96 Theil-Sen test) . Again the classical t-test performs poorly under Cauchy errors and generally best under normal errors, although at 6-,_2 = .100 under normal errors, the approximate Sievers-Scholz test had the highest pov/er. Summary of Conclusions The Monte Carlo results presented here generally concur with the asymptotic relative efficiencies (AREs) discussed in Section 2.4. Hence we draw the following conclusions about the power of the tests compared in the two line setting assuming common regression constants. First con- sider comparisons of the Sievers-Scholz, Hollander, and Rao-Gore methods, all three of which include exact confi- dence intervals for the slope difference g, „. The AREs and Monte Carlo results indicate superiority of the Sievers- Scholz approach when compared with the other two exact nonparametric procedures. Thus we recommend the use of the proposed application of the Sievers-Scholz approach over the Hollander and Rao-Gore methods. Mow consider comparisons of the Sievers-Scholz test. Sen's test, and the classical least squares theory t-test. As rank tests both the Sievers-Scholz and Sen test can be expected to be robust to gross errors, an advantage these two tests hold over the classical t-test. Although Sen's test has higher ARE than the Sievers-Scholz test under heavily tailed distributions, recall that Sen's test — ■T-.T ^v»> J . , ■-. ,i,, n y,— ,— >^<»^,^.,n^^Trs^-7-r**>— -gsr ,i , ii>, P ->. Tn .i m -W, »iA..- »; '>f^»wv*'-<«V 97 is distribution-free only asymptotically and does not include exact confidence intervals for the slope difference, ^1-2* ^^ addition, Monte Carlo studies by Smit (1979) and Lo, Simkin, and Worthley (1978) indicate the power of Sen's test under small sample sizes is quite conservative with respect to the corresponding classical least squares test. Hence the Sievers-Scholz method is preferred over Sen's technique when an exact, distribution-free test is desired that includes associated exact confidence intervals for the slope difference, &^_2- Although the classical test has the highest overall power when the underlying errors are normal, choice of the Sievers-Scholz test is appropriate when an exact test is desired that maintains good power over a range of distributions, including heavily tailed ones such as the Cauchy distribution. Our Monte Carlo comparisons show the Sievers-Scholz approach generally has greater power than the Theil-Sen technique when the regression constants are unequally spaced. When the approximate versions of these two tests are being used under large samples, the computations required by each are about the same. Hence in this case we ■ recommend the Sievers-Scholz technique over the Theil-Sen approach. When exact versions of the tests are being applied under sample sizes small enough to allow computation of the exact null distribution of the Sievers-Scholz statistic (8 or fewer observations per line) , we again 98 recommend the Sievers-Scholz technique over the Theil-Sen approach. For larger sample sizes when the exact Sievers- Scholz test is not feasible, we recomm.end the Theil-Sen test if an exact distribution-free test is desired. The portion of our Monte Carlo study that included the Sievers-Scholz approximation based on random samples of permutations has limited scope and produced mixed results. Thus, further investigation of this technique is required before a judg- ment of its usefulness can be made. CHAPTER THREE COMPARING THE SLOPES OF SEVERAL LINES 3.1 Introduction In this chapter we consider the case of several regres- sion lines, assuming coimnon regression constants for all lines. We examine the situation where one of the lines is a standard, or control, with which all other lines are com- pared. Suppose there are k lines and assume, without loss of generality, that the kth line is the standard. The specific topic of this chapter is the comparison of the slopes of the first k-1 lines with the slope of the kth line. As an example of the situation we are considering, suppose a dentist is examining some measure of strength for several different experimental amalgams used to fill cavi- ties in teeth. He may wish to compare the strength of each of these amalgams with the filling material he has commonly used in the past. His standard, then, is the commonly used material. Suppose he measures the strength of samples of each material, including the standard, after soaking in water for 2, 3, 4, and 5 days. The strength of an amalgam will generally decrease with increased soaking time. For each amalgam., consider a simple linear regression of the 99 100 response, strength, on the independent variable, soaking time. The slope of each regression line is a measure of the rate at which the amalgam's strength decreases v/ith increased soaking time. The methods suggested in this chapter apply when the dentist wishes to compare the slope of the line resulting from each experimental amalgam with the slope of the line resulting from the standard filling material. In the rest of this section we formally establish the linear model assumed and the null and alternative hypotheses under consideration. We propose a statistic applicable in this setting whose form follows as a generalization of the statistic T(b) (2.1.5), discussed in Chapter Two. In Section 3.2 we derive asym.ptotic distributions of the proposed statistic, and specify a test of the null hypo- thesis based on the statistic. In Section 3.3 we illustrate how our proposed statistic can be used to conduct exact tests of the null hypothesis. A competitor to the proposed test is defined and explained in Section 3.4. We close this chapter in Section 3,5 v/ith Pitman asymptotic relative efficiencies comparing the proposed test with its com.peti- tor . We now establish the linear model assumed throughout this chapter. Let Y . . = a, . + e .Y. . + E. . , f31M 101 where i = 1, 2, ..., k, j = 1, ..., N, and ^2 — ^2 — ' * ' - ^N * "*"" this model a . and g., i = 1, ,.., k, are unknown regression parameters, the x's are known regression constants, and the E's are mutually independent, unobservable random errors. We wish to test the null hypothesis -0* ^1 " ^2 " ••• " ^k (3.1.2) against the alternative -1* ^i ^ ^k ^°^ ^^ least (3.1.3) one i, i = 1 , . . . , k-1 . Thus the null hypothesis (3.1.2) is that all k lines are parallel. Note that this hypothesis can be expressed in terms of the slope differences, B- -B,, i=l, ..., k-1: So= ^1 - ^k = ^2 - \ = ••• = ^k-l - ^k = ° Define ? ^i: = ^ij -^kj' i - 1, ..., k-1, j - 1, ..., N. In terms of the underlying model parameters. 102 ^ij = ("i-°'k) + (^i-^k^^j + (^ij-\j)' (3.1.4) and so for fixed i, the differences Z. ., i = i n follow a simple (single line) linear regression model. The case k = 2 (i=l) was the topic of Chapter Two of this dissertation. Recall in that case our proposed statistic, T(b) (2.1.5), results from the use of a statistic originally suggested by Sievers (1978) and Scholz (1977) for the single line setting, when applied to the differences Z. ., i = l ..., N. Analogously, here we define for i = 1, ..., k-1 T. (b.) = (1/N) ZZ X sgn(Z. -Z . -b . x ), (3 15) X 1 ^^^ rs ^ 'is ir 1 rs ' k^.x.o) where b^ are constants, x^^ = x^ - x^, and sgn ( • ) is as used in (2.1.2). Note that T^ (b) is T(b) (2.1.5) of Chapter Two when only considering the ith and kth lines. Thus from the results in Section 2.3 and 2.5, T^(0) can be used to test ^Oi' ^i " ^k ag^i^st H^^: 3j_ ¥^ &^. Let 2 N - 2 ^x = .^ (^i"^) /N (3.1.6) j=l -J and 4, A = N'^x/^' (3.1.7) as in Chapter Two. 103 By Theorem 2.2.1 it follows that under H Oi' 'i ~ ^k ^"■'^ certain regularity conditions, the asymptotic (II^-) distri- bution of T.(0) T,A is standard normal. Note also that k-1 Q^0i=20' (3.1.8) that is, Hq holds if and only if h^^ holds for all i = 1, ..., k-1. The asymptotic* normality of each T.{0) under H X oi and the relationship C3.1.8) between the H^^ and H suggest a possible approach to constructing a statistic for testing Hq using the T^(b^), i = l, ..., k-1: Let T = -1 ^T,A T^{0) T2{0) Vl(0) (3.1.9) Suppose we can show that under H^ , the asymptotic distribu- tion of T is multivariate normal with mean vector (zeros) and nonsingular (k-1) by (k-1) variance-covariance m.atrix 104 i;.. Denote, the inverse of |; by | . Then it follows from results in Serf ling (1980, p. 25, 128) that the asymptotic (N-^°°) distribution of S = T ^"^ T (3.1.10) under H^ is central chi-squared with k-1 degrees of freedom. This asymptotic distribution could be used to construct an approximate level a test of H^ based on S. Our purpose here is to motivate the use of S (3.1.10), a suitable quadratic form in the elements of T, to test H . In the next section it will be shown that S does have an asymptotic chi-squared distribution under given conditions. The specific form of f and the test of H^ (3.1.2) against H^ (3.1.3) will be given, along with the distribution of S under a certain sequence of alternatives. 3.2 Asymptotic Theory and a Proposed Test In this section we develop the necessary asymptotic theory to derive a test of H (3.1.2) versus H (3.1.3) based on a suitable quadratic form in the elements of T (3.1.9). We also give the asymptotic distribution of T under a certain sequence of alternatives. This distribution will be needed in Section 3.5, when deriving asymptotic relative efficiencies. We first give some notation that will be needed in this section and the rest of Chapter Three. Some notation from "l'"*»W^VU'»i»*,*:t^.-i.«» -I 105 i Chapter Two will be repeated here, for the convenience of the reader. Basic notation . Let ^ indicate convergence in proba- bility and _^ indicate convergence in distribution. Let 2 N - 2 a^ = Z (x.-x) /N, (3.2.1) j=l -" 2 N - 2 2 ""t A " ^ (x.-x) /3 = Na;/3, (3.2,2) ' i=l -^ s-1 ^"s " .^ ^^s"^j^ ' (3.2.3) N X • = Z (x.-x ) , and so (3.2.4) j=r+l -■ ^ x.j - Xj. = N(Xj-x) (3.2.5) by simple algebra. If the event A occurs with probability one we write A, w.p. 1. We adopt the 0(«), o(«), (•), o (•), and ^ notations as in Serfling (1980, p. 1, 8-9). Matrix notation . Some matrix notation v/ill be needed, A vector of size n is considered to be an n by 1 matrix (a column vector) . Two such vectors that will be needed are 106 X = (3.2.6) n a vector of constants used in forming linear combinations, and = (3.2.7) the vector of zeros. The dimensions will be implied by the context. Also let ^k = 1 • • • 1 • • • 1 • • • ... 1 (3.2.8) the identity matrix of order k, and ^k = 1 1 1 .. . 1 1 1 1 ... 1 1 1 1 ... 1 (3.2.9) 107 the k by k matrix whose elements are all one (1) . When multiplication of a matrix by a scalar is indicated, all elements of the m.atrix are multiplied by the scalar, i.e.. 5J3 = 5 5 5 5 5 5 5 5 5 The inverse of a matrix, if it exists, is indicated by a superscript -1, i.e., I"""- | = I^^. Finally, X' indicates the transpose of the vector _A.' Distributional notation . We assxime the underlying error distribution of all lines is the same, with cdf F and pdf f. Let E . , j = 1, 2, ..., designate independent and identically distributed random variables with cdf F. G and g denote, respectively, the cdf and pdf of E - E . The cdf and pdf of E^ + E2 - E- are denoted by M and m, respectively. Let A(F) = P{E^<E2+E3-E^, E^<E2+Eg-E^} - 1/4 (3.2.10) and note that A (F) may also be written A{F) = /[l-M(e)]''dF(e) - 1/4 (3.2.11) if the integral exists. Let 108 A* = 12A(F), (3.2.12) and for general pdf q let Kq) = /q^(x)dx, (3.2.13) provided the integral exists. For the multivariate normal distribution of dimension k with mean vector u and covariance matrix ^ we use the notation N^^ii'l)- (3.2.14) For the chi-squared distribution with k degrees of freedom and noncentrality parameter 6 we use the notation 2 X (<S)- (3.2.15) Hypotheses . The following hypotheses will be referred to frequently: -0' ^1 " ^2 " ••• " ^k (3.2.16) -1' ^i ^ ^k ^°^ ^^ least one i, (3.2.17) i = 1, . . . , k-1 ^Oi= ^i = ^k (3.2.18) ^li= ^i ^ ^k (3.2.19) 109 Hj,: B. = Bj^ + ((../(Nk)^a^) (3.2.20) For expectation under H^ we use E . Conditions. The following set of conditions will be used in the statements of the theorems in this section: ^ - 2 2 I. I (x.-x) = Na -> ~ as N ^ cc. j = l ^ ^ II. max (x.-x)^ max (x.-x)^ l^J^N ^ 1^<N ^ ^iq^^ - — 2 ->• as N > 2 (x.-x) 2 Na^ j=l ^ III. G is continuous. IV. G has a square integrable absolutely continuous density g. We now prove that the asymptotic distribution of T (3.1.9) is multivariate normal, as conjectured in the previous section. We first state and prove two lemmas. Lemma 3.2.1; Let {X^, n ^ 1 } be independent bounded random variables with X nl 1 ^n' ^'P* ^ for n ^ 1. (3.2.21) — - •!*— ^ ■ii»»,'>j1i»|t^— - ■^f^.. 110 ^ 2 ^ 9 Let S^ = IX. and s = I E (X -EX ) ^ n > 1, and assxme 3=1 J " j=i J J 2 s^^ -> CO as n -^ CO. If the bounds B satisfy ^n = °^^n^ (3.2.22) then S - ES i ^ . N(0,1). (3.2.23) n Proof of lemma ; ^^^ ^n = ^n ~ ^^n- "^^^^ ^<^n^ = °' ^^^ ^^ ^ j!l^^''j"'=''j^^= -Ei^^i^- (3.2.24) In view of Liapounov's corollary (Chow and Tiecher, 1978, p. 293) to the Lindeberg-Feller central limit theorem, the result (3.2,23) follows if we can show the following conditions hold: For some 6 > , 1) E|Yj2+fi <„,„>! (3.2.25) 2) ,^_^E|Yj| ^ = o^^n"^"^) as n -^ -. (3.2.26) Ill We will show (3.2.25) and (3.2.26) hold for 6=1. From (3.2.24) we see eIy P < E(2B )^ = 8B"^ < «, n > 1, and so n ' — n n — (3.2.25) holds for 6=1. Now z e|y.|^ i: e(|y |2|y I) j=l -' ^ 2^ So 3 3 n n Z^E(|Yj| 2B.) (^^^^ (3.2.21)) n 2 max B-. Z E( |y. | ) n 2 max B . s (from (3.2.24) ) . n E E I Y . I 2 max B . izl_i_ < _IlJin_2 , (3.2.27) If V7e can show ■ umnii 'i L i ii i iL-j ii jiM«j i i. i M ii j i» . i m f tf « »j i n ■! Il l »i »ii«a »» ir 112 2 max B . n (3.2,28) as n ^ «, then the Liapounov condition (3.2.26) holds for 5 = 1 and the proof is complete. In the following let m be an integer such that m + 1 < n: max B max B . max B . l^i^n ^ l:£i^m ^ _^ m+l^i^n ^ n n n max B . ^ l^j^m -* , , , . 1 s ^ ^^^ (B./s.). n m+l<j<n -' -^ (3.2.29) Hold m fixed and let n -> «> in (3.2.29) max B . lira i£^lIL_ < + sup (B./s.). n-*-" n j>m+l -' -^ (3.2.30) Since the left-hand side of (3.2.30) does not involve m, we let m -> »: max B . lim — !££ — = lim sup (B./s.) = 0, n->-<» n m>°° j>m+l -' ^ (3.2.31) where the right-hand side of (3.2.31) equals zero because Bj = o(Sj) (3.2.22). So (3-2.28) is true and the proof is complete. 113 Leirana 3.2.2 ; Define for i = 1, , k-1. N T. (0) ,^ E [T (0)|E -E ], = 1 -0 ^ ^3 ^3 (3.2.32) where T^(0) is T^(.b^) in (3.1.5) with b. set to zero. Let T = a -1 T,A T^(0) T2(0) (3.2.33) L Tj,_,(0) Under Iq* ^1 " ^2 " * III, - 3j^, assuming conditions I, II, and i t \^i^9.4) (3.2.34) as N ->■ "^ where t = Vi + A* (\.r\-i^' (3.2.35) and A* is given by (3.2.12). 114 Proof of lemma : From (3.1.4) it follows that under H^ , Z. -Z. =E. -E, -E. +E, , IS ir IS ks ir kr Then E„ [T. (0) E. .-E, .] H^ 1 ' ij kj-" = E„ [4 SZx sgn(Z. -Z . ) Ie. .-E, .] H„'N rs ^ 'is ir' ' 11 ki-' —0 r<s J -I E[rz izx sgn(E. -E, -E. +E, ) Ie. .-E, .] N rs ^ 'is ks ir kr' ' ig k:]-" = ^^ZZx^3E[sgn(E.3-E^3-E.^+E^^) |E. ^-Ej^.] . (3.2.36) For notational simplicity let D = E. .- E, . . Then examining 13 K^] E[sgn(E.^-E^^-E.^+Ej^P |e. j-E,^..D] , (3.2.37) note the following: 1) If s = j then D is independent of E. - E, and so (3.2.37) equals 115 P{D>E, -E, } - P{D<E. -E, } ir kr xr kr"^ = G(D) - {l-G(D)) = 2G(D) - 1. (3.2.38) 2) Similarly, if r = j, then (3.2.37) equals 1 - 2G{D) . (3.2.39) 3) If r 7^ j and s ?^ j , then D is independent of ^is " ^ks " ^ir ■*■ ^kr ^^'^ ^° (3.2.37) equals E[sgn(E.^-Ej^^-E.^+Ej^p]. (3.2.40) Now E^^ - Ej^g - E^ + E, is symmetric about zero so (3.2.40) equals zero. Recall the notation x. (3.2.3) and x . (3.2.4). Applying 1), 2), and 3) above to (3.2.36) results in the following: %tT.(0)lE..-E3^.] = TT ZZx E[sgn(E. -E, -E . +E, ) Ie. .-E, .1 N _^ rs ^ ' IS ks ir kr' ' 13 k^ 116 2 J-1 N n[ r X (2G(D)-1) + r x(l-2G(D))] r=l J s=j+l -"^ |[{2G(D)-l)x_ + (l-2G(D))x..] ^(2G(D)-1) (x.^-x^.) = ^(2G(D)-l)N(x.-x) (from (3.2.5)) = (2G(D)-l)(x.-x). (3.2.41) Substitute (3.2.41) into (3.2.32): N T,(0) = fjE2^[Ti(0)|E..-E^.] N = T (2G(E -E )-l) (x.-x) N Z_^2(Xj-x)G(E^j-Ej^j). (3.2.42) We have now established the form of the T. (0), i = 1, k-1. Consider an arbitrary linear combination of the i\(0) -s. 117 k-1 . .f/i^i^O). (3.2.43) Our basic approach to proving this lemma is to show the asymptotic normality of this linear combination (3.2.43). Let k-1 Uj = 2(x.-5)_Z A.G(E -Ej^n. (3.2.44) i=l From (3.2.42) , k-1 . k-1 N J^A,T,(0, - ,f^VJ^2(x.-5,G(E,.-E,., N _ k-1 = Z [2 (x.-x) Z A.G(E. .-E, .) ] j=l 3 i^i 1 13 k:' J N ■ l^^r (3.2.45) Note that U^ and U^ , are independent, j 7^ j ' . Before deriving the mean and variance of U , observe the following: 1) The cdf of E^j - E^. is G (i<k) . Let g"^ be an inverse of G, i.e., G"^(G(t)) = t , <_ t ^ 1 . G~^ exists by continuity of G (condition III) . Then the cdf of G(E^j-E^.) is given by -.--.-.-.^^1.3= 118 P{G(E..-Ej^.)<t} Pi'E^^-E^.<C~^ it) } G(G ^(t)) = t, < t < 1. Thus G(E^j-Ej^. ) is uniformly distributed over [0,1] . It follows that E[G(E. .-Ej^.)] = 1/2, (3.2.46) and E[G(E. .-Ej^.)-l/2]2 = 1/12, (3.2.47) the variance of a uniform [0,1] random variable. 2) Recalling the definition of A (F) (3.2.10), it follows that A(F) + - = P{E2<E^+E^-E2, E2<E3+Eg-E^} = E tP{E5-E4<E^-E2 ,E^-Eg<E3-E2 | E^=e^,E^=e^,E^=e^}] 119 = E[P{E5-E^<e^-e2,E^-E^<e3-e2|E^=e^,E2=e2,E3=e3}] PiE^-E^<e^-e^\E^=e^,E^=e^,E^=e^}] = E[G(E^-E2)G(E3-E2)] , and so using (3.2.46) we have E[{G(E^-E2)-l/2) (G(E3-E2)-l/2)] = A (F) . (3.2.48) Then k-1 EU. = E[2(x.-x) Z A.G(E. .-E, .)] = 2(x.-5)_Z^A.E[G(E..-E3^.)] k-1 (Xj-x)_z X^ (from (3.2.46)), (3.2.49) and 120 Var(U.) = E[U.-EU.] ^ k-1 ^ = E[(2(x-x) Z X G(E -E ))-((x,-x) I A.)] -» i=i -^ -^J ^-J J j__2 1 k-1 = E[2{x.-x) I X. (G(E.j-Ej^.)-l/2)]2 - 2 '"-I 2 = 4{x -X) { I X^E[G(E.,-E, .)-l/2] + 2,2Z^A.X.,E[{G(E. .-Ej^.)-l/2) (G (E . , ,-Ej^ . ) -1/2) ] } - 2 ^-1 P = 4(x -X) [(1/12) E A^ + A(F)2 ZI A.A.,], (3.2.50) i=l i<i' ^ ^ where (3.2.47) and (3.2.48) were applied to arrive at the final expression (3.2.50). Now we show that the conditions of Lemma 3.2.1 apply to the U.. First note lu^l = I2(x^-5)_I^A.G(E.^-E^^)| < 2 max ix -x| z | A | , w.p. 1 l<j<N -' i=i ^ 121 k-1 Let B = 2 max |x -x| z |^ |, and then l<j<N -' i=l ^ U^l 1 B^, w.p. 1, N > 1, (3.2.51) SO (3.2.21) of Lemma 3.2.1 is satisfied. Applying (3.2.50), 2 ^ if we let s = z Var(U.), then j=l ^ 2 ^ - 2 ^-^7 s^ = 4 S (X -x)^[(l/12) Z Af+A(F)2 ZZ A.A.,] j=l ^ i=l ^ i<i' 1 ^ N _ k-1 _ = (1/3) Z (x,-x)^[ Z Xf+2A* ZZ \.X.,]. (3 2 52) j=l ^ i=l ^ i<i- 1 1 U.^.i^) For later reference we note: 4 = Varjj^(ZA.T.(0)), (3.2.53) and taking A. = 1, a^, =0, ± f x' the form of the null variance of T^(0) follows from (3.2.52): r.2 2 Var_^_^(T.(0)) = (1/3) Z^(x_.-x)^ = a^^^. (3.2.54) 122 Now examine B„/s : N N k-1 max |x.-x|2 Z \x. B^/Sm = l^i^N -J i=l ^ {(1/3) Z (X -x)'^[ Z X^+2A* EZ X.X.,]}^ j=l ^ i=l ^ i<i' 1 1 - as N - " (3.2.55) by condition II, and so (3.2.22) of Lemma 3.2.1 holds. We note from (3.2.49) , N N k-1 2 EU. = Z (x.-x) Z A. = j=l ^ j=l 3 ,i=i 1 Applying Lemma 3.2.1, N N 2 U . - Z EU. ^ 1=1 ^ j=i : d -^ i ^-^^— ^ N(0,1) N as N ->■ » , N Z U. ^ -^i > N(0,1) (from (3.2.56)), (3.2.56) 123 :> k-1 . Z A.T. (0) i=l d => N(0,1) (see(3.2.45) ) , N _1 k-1 . a Z XT. (0) i=l d ^I ;N(0,1), (3.2.57) as N ->• " Let A be a vector as in (3.2.6) and recall the definition of the matrix | (3.2.35). From (3.2.52) the following expression results: ,-2 2 , -2 N - 2 ''t^A^N = f^T,A(^/3) Z (X -x)^]A'tX j=l -^ = ^^tVt,a^A'^A = A'iA. (3.2.58) Using (3.2.58) in (3.2.57) we see =%fN(0,l) (3.2.59) A'l d ix'tx) as N -^ CO, where T is given by (3.2.33). The result (3.2.59) essentially shows that any linear combination of the 124 elements of T converges in distribution to a random variable with the distribution of the same linear combination of a \_l(0.4) random variable. By a theorem in Serfling (1980, p. 18) this is equivalent to showing T has an asymptotic (N->") Nj^_;l^^'^^ distribution, and the proof of Lemma 3.2.2 is complete. Theorem 3.2.3: Under H^ : 3^ = 33 = . . . = 6j^, assuming conditions I, li, and III, d "k-l^-' + '' (3.2.60) T " N,,_^(0,t) as N -. -, where T is given by (3.1.9) and t is given by (3.2.35). Proof ; Consider f or i = 1 , ..., k-1, E„ (T. (0)-T. (0))^, -0 ^ Where T^(0) is given by (3.2.32). T^ (0) is called the projection of T. (0) on the Z..=E..-E i=l n under H^ . By a lemma due to Hajek and appearing in Serfling (1980, p. 300), it follows from the form of T. (0) that E (T.(O)-T. (0))2 = var^ (T.(0)) -Var^ (T, (0)), u —0 —0 (3.2.61) 125 1, ..., k-1. From (3.2.54), Var^^ (T^(0)) = aj^^, (3.2.62; and from Theorem 2.2.1 applied to this situation. Varjj (T^(0)) = a^^^ + o^. (3.2.63) Substituting (3.2,62) and (3.2.63) into (3.2.61) yields E (T.(0)-T.(0))2 = ia^^-^ol) - a^^ = o^ ■■> lim a \e (T. (0)-T (0))2 = lim a^ o^ N - 2 2 (Xj-X)^ j=l ^ = 0, (3.2.64) for i = 1 , — , k-1 . Then it follows that crT^^^T^(O) - T^(0)) I (3.2.65) as N ^ ", i = 1, ..., k-1. This implies 126 _l k-1 k-1 ^T^A^^^i^i^O) - ^ ^iTi(O)) P (3.2.66) 1=1 i=i -^ -^ as N ^ oo. Using the matrix notation (3.2.6), (3.2.33), and (3.1.9), we can express (3.2.66) as follows: X'T - X'T P (3.2.67) as N ^ 00. Thus from Slutzky's theorem (Randies and Wolfe, 1979, p. 424), (3.2.67) implies that if X' T has a limiting distribution then X'T has this same limiting distribution. In Lemma 3.2.2 we saw that under H^ , assuming conditions I, II, and III, X ' T ~ ~ t: t N(0,l) ix'tx) as N -> 00 (from (3.2.59)), and so k . N(0,1) (3.2.68) iX'iX) as N ^ 00. As argued in Lemma 3.2.2, this is equivalent (Serf ling, 1980, p. 18) to showing T has an asymptotic (N->oo) N]^-l(£'t) distribution, and the proof of Theorem 3.2.3 is complete. 127 Notice that | (3.2.35) has the form abb b a b b b a b b b . b b b (3.2.69) with a = 1 and b = a* = 12A (F) . Then by Theorem 8.3.4 in Graybill (1969, p. 171), | has an inverse if and only if (iff) 1 7^ 12A(F) , (3.2.70) and 1 7^ -(k-2)12A(F) . (3.2.71) Mann and Pirie (1982) give the following bounds for A (F) : 1/36 < A(F) < 1/24. (3.2.72) The bounds (3.2.72) hold for all continuous distributions. Condition (3.2.70) fails iff a (F) = 1/12, which, in view of (3.2.72), is not possible for continuous F. Condition (3.2.71) fails iff A(F) = -[12(k-2)]"^ and k ^ 3. By (3.2.72) we see that A (F) cannot be negative, and so 128 (3.2.71) holds for continuous F. Hence for continuous F, ^ has an inverse, i~ . Again appealing to Graybill's (1969, p. 171) theorem, the inverse of | is given by t = riA^^^k-l ~ l-(k-2)A* "^k-l^ • (3.2.73) Consider the quadratic form S = T'f^T, (3.2.74) T as in (3.1.9). Using (3.2.73) and some algebra, S can be expressed: k-1 ^ k-1 (l+(k-2)A*) I (T^(0))-A*( S T^(0))^ S= ^^ ^=^^ . (3.2.75) a^^^a-h*) (l+(k-2)A*) The following gives the asymptotic distribution of S under Theorem 3.2.4 : Under H^ : 3^ = g^ = . . . = g^, assuming conditions I, II, and III, S ^ X^_i (0) (3.2.76) as N -» «, where S is given by (3.2.74) 129 Proof ; From Theorem 3.2.3, 1 J X as N -»- ", under H^ , where X is a random vector having the N]^_l(£4) distribution. Then from Corollary 1.7 in Serf ling (1980, p. 25), it follows that S = T' Z~"^T J X' t~'''X as N ^- ", under Hq . Applying a theorem in Serf ling (1980, p. 128) it follows that X'T'^X 2 has the Xj^_j^(0) distribution, and the proof is complete. The result of Theorem 3.2.4 suggests a test of -0* ^1 " ^2 " ••• " ^k ^^- -1 (3.2.17). Let X^_^ be the cdf of the central chi-squared distribution with k-1 degrees of freedom. For < a < 1, let ^^_^ ^ be the upper 100 (1-a) percentile, that is X(iJ;j^_^ ^) = i _ a. Then from Theorem 3.2.4, the test based on S that rejects H_ (3.2.16) if ^■^'^k-l,a (3.2.77) is an approximate level a test of H against H (3.2.17) 130 Of course the parameter A (F) depends on the underlying distribution and so the value of S depends on the underlying distribution. Consistent estimates of A (F) are discussed by Mann and Pirie (1982), however such estimates are often tedious to compute. Consider Table 24, which is adapted from Mann and Pirie (1982) and shows the value of A (F) to four decimal places for several common error distributions. Table 24. Values of A (F) . F Uniform Normal Logistic Exponential Cauchy Max. Min. A(F) .0409 .0402 .0398 .0394 .0379 .0417 .0228 Source: Adapted from Mann and Pirie (1982) . The last two columns of Table 24 give the maximum and minimum values of A (F) for continuous F, discussed previously (3.2.72). We see the range of possible values for A(F) is about 0.02, and all of the values listed remain within 0.01 of the maximiom. The range of possible values for A* = 12A(F) is (l/2)-(l/3) = 1/6 or about 0.17 for continuous F (see (3.2.72)). In view of this small range we suggest a test based on S which replaces A* by a value A* c' chosen to yield a conservative test. We now indicate how to choose the value of A*: c 131 Let k-1 T - I T,(0)/(k-l) (3.2.78) i=l ^ 2 ^-^ - 2 s^ = ^ (T. (0)-T)^/(k-l) , and (3.2.79) i=l 2 2-2 c^ = s^/ (T) . (3.2.80) Then using these definitions in the expression for S (3.2.75) we have 2 S = S(A*) = [(k-DT^/cT^ ] [-!t__ + 1 _ j^ 1-A l+(k-2)A (3.2.81) Examining the derivative of S(A*) with respect to A*, S' (A*) , it can be shown that S' (A*) is >, = ,< (3.2.82) (k-2)^ - c^ as A* is >, = ,< — (k-2)Cy + (k-2)^ Let S(l/3), S(l/2) be the value of S(A*) when A* is set at it lower and upper limits of 1/3 and 1/2, respectively. 132 Let r 1/3 if cl > 1"^ T 2 A* =/ ^^^ " ""t , if kzi < ^2 ^ 4(k-2) "" \ (k-2)c^ + {k-2)'2 k^ - T - (3^^^ J 2 1/2 if c^ < ^ . (3.2.83) Using (3.2.82) and the upper and lower bounds of A* it follows from some routine analysis that min S(A*) = S(A*). (3.2.84) (1/3) < A* £ (1/2) ^ . Then if we define S(A*) to be the value of S (3.2.75) using * A* = A , it follows from (3.2.84) that the test of H„ (3.2.16) which rejects in favor of H (3.2.17) when ^(^^ ^ Vl,a (3.2.85) will be conservative. Its true level will be less than or equal to the nominal level a. However in view of the small range of A* under continuous F, the degree of conservatism of the test (3.2.85) should be slight. Suppose 0)^^, 0^2 ..., 0)^ are given constants such that ojj^ = 0, and consider the sequence of alternatives. 133 ^N = 'i = 3^^ + to^/{Nk) ^a^, (3.2.86) 1 X y • * • f K I Let (0 = OJ- 0). ^k-1 (3.2.87) The following theorems give the asymptotic distribution of T (3.1.9) and S (3.2.74) under H„. Theorem 3.2.5. Let ^, o), and T be given by (3.2.35), (3.2.87), and (3.1.9), respectively. Under H (3.2.86), assuming conditions I-IV, T t Nj^_i(2(3^)I(g)k-\,|) (3.2.88) as N ^ «>. Proof ; Let 0)^^ = oj^/N^^a^, i = of alternatives is H„: g. = — N 1 1, ..., k. Then the sequence ^k "*" "iN' ^ ^ ^' •••/ k. Let 134 ^^-i^N) = ^t]a Vi (-"k-1 n) (3.2.89) As noted in Chapter Two, the limiting distribution of T = T{_0) under H^ is the same as the limiting distribution °f I(~%) under H . = 3, . So assume H- and "1 ^2 show that T(-i;Jfj) has the desired limiting distribution. Consider an arbitrary linear combination of the T. (-u. ) 1 iN ' k-1 i=l ^ ^ iN' (3.2.90) Using the matrix notation (3.2.89) and (3.2.6), we can express the linear combination (3.2.90) as -t,aA't(-%) (3.2.91) Our basic approach to proving Theorem 3.2.5 is to show this linear combination has an asymptotic distribution that is univariate normal. We first state and prove two lemmas. Lemma 3.2.6 : Assume H conditions I-IV. Then ^1 = = B, and 135 %f.!^^i^i(-^N^/^T,A^ - 2(3'^)I(g)k^^E^A.co. (3.2.92) as N ->- «> . Proof of lemma ; Applying Lemma 2.2.3, we have in the notation of this chapter. Ejj . fT^(-aj^j^)/a^^^] ^ 2 (3^) I (g) k^u^ (3.2.93) asN-><=°, i = l, ..., k-1, H_ . given by (3.2.18). Since H-^ implies Hq., i = 1, ..., k-1 (see (3.1.8)), the result (3.2.92) follows immediately. Lemma 3.2.7 ; Assume H- ; B- = Bj = ••• = S, and condi- tions I-IV. Then .^^i^i(-'^iN) - t.^^i^i^O) -^ V(.^^iTi(-'OiN))] 1=1 1=1 —0 1=1 ''t,a (3.2.94) as N ->■ ". Proof of lemma; Applying Lemma 2.2.4, we have in the notation of this chapter. 136 ^i^-^N^ - f^i^O)-^ V.(Ti (--,,))] Ol p ^ : (3.2.95) T,A asN^co, i = i^ ._^ k_l^ Since H. implies H„ . , —0 '^ 1 i = 1, ..., k-1 (see (3.1.8)), the result (3.2.94) follows immediately. Proof of Theorem 3.2.5 (continued) ; Under H^^ and conditions I, II, and III, Theorem 3.2.3 (see (3.2.68) ) showed X'T J N(0,X'|X) (3.2.96) as N -> CO, where ^ is given by (3.2.35). The result of lemma 3.2.6 in matrix notation is Ejj [A'I(-%)] ^ 2(3'2)I(g)A'iil (3.2.97) as N -> =0. Using (3.2.96), (3.2.97) and Slutzky's Theorem (Randies and Wolfe, 1979, p. 424), it follows that X'T + Ejj U'T(-co^)] J N(2(3'2)I(g)A'iil'i'lA) (3.2.98) as N ^ =°. The result of lemma 3.2.7 in matrix notation is ^'T(-aij^.) - U'T + Eg (A'T(-aij^))] ^ (3.2.99) 137 as N ^ 00. So using (3.2.98) and (3.2.99), a second application of Slutzky's theorem shoxvs ^'T(-a)jj) t N(2(3^)I(g)A'i!l/ A'ID (3.2.100) as N ^ ", under H^ . Put another way, (3.2.100) reveals that the asymptotic distribution of A'K-w^) under H is the distribution of ^'X, where X has the Nj^_^ (2 (3^) I (g) k\, ^) distribution. Hence by a theorem in Serfling (1980, p. 18) this is equivalent to showing T(-u ) converges in distribu- tion to this same multivariate normal distribution. Hence by our remarks at the beginning of the proof of Theorem 3.2.5, the proof is complete. Theorem 3.2.8 ; Let S be given by (3.2.74). Under H (3.2.86), assuming conditions I-IV, k-1 , k-1 (l+(k-2)A*) E a3T-A*( Z to . ) ^ S t xj_i([12l2(g)][ k(l-A*Ml+'(k-2)iM ^ ^) (3.2.101) as N -»■«>, A* given by (3.2.12) Proof: From Theorem 3.2.5, under H„, — N 1 J X 138 as N -> CO, where X has the Mj^_^ (2 (3^) I (g) k^oj , t ) distribution, Then from Corollary 1.7 in Serfling (1980, p. 25), it follows that S = IT^T f X't ^X (3.2.102) as N > 00, under H^^. Applying a theorem in Serfling (1980, p. 128) , X' t~-^X has the Xj^_^(12l (g)k~^m'i~^w,) (3.2.103) distribution. The noncentrality parameter in (3.2.101) is just the one in (3.2.103) expressed in terms of the elements of 0) (3.2.87) and ^""^ (3.2.73). 3.3 An Exact Test The statistic S (3.2.75) can be utilized to construct an exact test of H^ (3.2.16) against H^ (3.2.17). The idea is similar to that used for exact tests based on T(0), described in Chapter Two. After first establishing some notation, we will review the case of k=2 lines. We then extend this method to the situation of several lines. Recall the definition Z. . = Y. . - Y, . i: ID kj (a.-aj^) + (6-e^)x, + (E.j-E^j), (3.3.1) 139 where Y^^ follow the linear model (3.1.1), i = 1, ..., k-1, 3=1, ..., N. Let R . . be the rank of Z . , amora (Z^^, ..., Zj^^}, the Z's resulting froir, the observations at the ith and kth lines. Consider the vectors R. -3 R Ij Z 2j R. k-1 j J (3.3.2) j - 1, ..., N, and the matrix composed of these vectors, R = [R^ R2 ... P^] . (3.3.3) The following representation of T. (0) in terms of the R^ 1 i j ' Z •••/ Rj,j^T results from (2.5.1): ^ 7 T.(0) = (2/N) 2 Rf X . - (N+l)x j = l ^^ -' (3.3.4) When k=2, R has dimensions 1 by N. In this case -0 " ^01* ^1 " ^2 ^"^ under H^^ it follows from (3.3.1) that the Z^^, ..., z^^^^ are i.i.d. and hence the distribution 140 of R is uniform over the N! permutations of its elements, the integers 1, ..., N. Thus, as discussed in Chapter Two, the null (6^=62) distribution of T. (0) can be determined by using (3.3.4) to compute the N! equally likely values of T.(0). Now note that when k > 2, the matrix R is (k-1) by N and the elements within the jth column, R., are dependent, since their values all rely, in part, on the value of Y . . ik To take care of these dependencies, we condition on the values of the columns of R. That is, under H : ^1 ^ ^2 "^ •*• ^ ^k' ^^^ distribution of R conditional on the observed values of R^ , j=l, ..., n, is uniform over the N! permutations of its columns. The conditional null distribution of R just described can be used to conduct an exact test of H against H using S(A^), defined after (3.2.84). The N! values of T (3.1.9) resulting from this conditional null distribution of R would be used to compute N! values of S(A*), each occurring under Hq v/ith conditional probability 1/N! . Suppose, for an observed set of data, s is one of these values such that P{S(A*)^s} = a. where the probability is evaluated using the conditional null distribution. Then, for this data set, the test which rejects H^ (3.2.16) in favor of H, (3.2.17) if 141 S(A ) ^ s is an exact, conditional level a test. Note that the degree of difficulty of computation of the given permutation distribution depends only on the number of regression constants per line, N, and not on the number of lines, k. o In view of this fact and the results discussed in Chapter Two, we found the exact test of H presented here to be computationally feasible with the use of a computer whenever N _< 8 . 3.4 A Com.petitor to the Proposed Test The statistic S (3.2.74) resulted from an obvious generalization of the proposed application of the Sievers- Scholz statistic T(0) for use in comparing the slopes of two regression lines. Clearly one could consider such gen- eralizations based on the other statistics for testing — 0* ^1-2 ~ ^1 ~ ^2 "^ '^' P^ssented in Chapter Two. In view of the fare's reported in the previous chapter, we shall confine our attention to the statistic defined by Sen (1969) , keeping in mind that exact tests based on a permu- tation distribution are not possible for Sen's test statistic. We would anticipate that, in terms of PARE, Sen's test is likely to be better than the test based on S. Recall that the test based on S was designed to test lio= ^1 = ^2 = ... = 3, (3.4.1) 142 against -1* ^i "^ ^k ^^^ ^'^ least one i, f3.4.2) i = 1, . . . , k-1. Sen proposed his statistic for use in testing H against the more general alternative, ^1 + ' ^ir •••/ 3j^ are not all equal, (3.4.3) and hence it would not be appropriate to compare his test against the test based on S. Instead we modify Sen's statistic to be specifically sensitive to the alternative H- (3.4.2). A test based on the modified statistic follows from the results of Sen. In the next section we compare this test with the test based on S . We note that the test based on Sen's results will not require common regression constants for all lines as required by the test based on S. However, as in Chapter Two, we will assume this is true to ease the notation and also to facilitate the comparisons in the next section. Thus we assume the basic linear model (3.1.1). The notation defined in Chapter Two when introducing Sen's statistic in the two line setting will be repeated here for convenience of the reader. Let (j) (u) be an absolutely continuous and nondecreasing function of u: < u < 1, and assume that <^ {u) is square integrable over (0,1). Let U,,. < U,„. < ... < U ,,,, be the (-L) (^) (N) 143 iii order statistics of a sample of size N from a uniform [0,1] distribution. Then define the following scores: Ej = E[MU^.j)], (3.4.4) or E. = Mj/N+1), (3.4.5) j = 1 , . . . , N. Define and 1 (j)* = / (J) (u)du (3.4.6) A^ = / (()2(u)du - (<j.*)2, (3.4.7) • and consider the statistics '' U V = [ Z (X-;-^)E ]/(AN^a ) (3.4.8) j=l ^ ^ij ^ i - 1, 2, ..., k, where R . . is the rank of Y.. among Y.,. ID 2.J ^ il ' ^12' •••' ''^iN' ^^® observations of the ith line. Recall that statistics such as (3.4.8) are used in the single line setting to test hypotheses about the slope. 144 Assume F, the underlying error cdf of the assumed model (3.1.1), is absolutely continuous and CO IMF) = / [|^]2dF(x) < <.. (3.4.9) Then we again reserve the symbol ¥ (u) for the optimal score function: T(u) = -^IJlImI , < u < 1. (3.4.10) f(F ^(u)) It can be shown, 1 / 'y (u) du = , and 1 2 / Y (u)du = I* (F) We also define P(^,<t>) = [/ 'i'{u)<f (u)du]/[A^I*(F)]^, (3.4 11) which can be regarded as a measure of the correlation between the chosen score function cp and the optimal one {"H) 145 I for the error distribution being considered. The expres- sions p(T,(j)) (3.4.11) and I* (F) (3.4.9) Vi7ill appear in the development of the statistic based on Sen's (1969) work. We now define Sen's statistic in the case of several regression lines and then give our modification. Let V^(y^+bx) (3.4.12) denote the value of V. (3.3.7) based on Y . , + bx, , Y.^ + 1 il 1' i2 bx^, ..., Y.^ + bx . Define V = r V./k. i=l Assuming H^ , let 3* denote the Hodge s-Lehmann estimate of the common slope of all k lines based on V. Define ^i " ^i(Xi-^*2i) ' (3.4.13) i = 1, 2, ..., k. Then Sen proposed the statistic k ^2 L = Z V. (3.4.14) i=l ^ to test Hq (3.4.1) against H^_^ (3.4.3). The statistic L is a quadratic form in the V. , i = 1 , . . . , k. We gave an 146 intuitive motivation for the forin of L in Chapter Tvro. We now define the modification of L. Let 3| denote the Hodges-Lehmann estimate of 6 . based on V^ (3.4.8), that is, based only on the observations from the ith line. Let Vj_ = V. (Y^-B*x)/ (3.4.15) i = 1, ..., k, and define A k - L = Z (V.-V)2, (3.4.16) i=l ^ - k . where V = Z V./k. Since we are interested in rejecting H. i=l ^ only when the slope of one of the k-1 lines differs from the slope of the kth line, we propose transforming the observa- tions by using the estimate of the slope of the kth line, 3*, rather than the pooled slope estimate, 6*. We again consider a quadratic form, now in the V., i = 1, ..., k. Thus we have made a reasonable modification in the form of Sen's original statistic. The basic asymptotic theory still holds, and proofs leading to the specific form of a test of H_ against H- based on L will be given. We now state the basic set of assumptions required for the results in this section. These are the assumptions Sen "■^-1 TTf^-nrrar-r-. 147 gives, applied to the setting of common regression constants: ^ - 2 A. Z (x.-x) •> <» as N ^- <=°. j=l ^ max (x.-x) B. l^i^ N ^ — jj -^ as N ^ 2 (x.-x) 2 j=l ^ C. F is absolutely continuous with (3.4.9) I* (F) < 00, D. The score function <f) (u) is an absolutely continuous and non-decreasing function of u, < u < 1, that is square integrable over (0,1) . We note that conditions A and B are identical to conditions I and II, respectively, defined previously. All four of these conditions will be referred to jointly in the follow- ing proofs as conditions A-D. Finally, we state again the sequence of alternatives we consider: n^: 3. = 63^ + co./N^k^a^ (3.4.17) -L. X f • * • f }\. f where 0)^ = 0. Most of the following lemmas are proved in the work of Sen (196 9) . ■^t-S-,V---W^- .,?Ki^ii 148 Lemma 3.4.1 ; Under H^ (3.4.17) and the conditions A-D, N^a^(B^-3|) I = Op(l) (3.4.18) and N^0^(6^-S*)| = (1) (3.4.19) as N •» «>, i = 1, . . . , k. Proof of leiruna: The result (3.4.18) is equation (3.20) in Sen (1969, p. 1675) , when using the notation of this section. Result (3.4.19) follows from (3.4.18): Under E^: ^^ = &^ + <^i/N^k^ax, |N\(6.-e*) = lN^o^^(6k-3p + a)^/k'^ - I^^'^x^^k-^k^ 1 + Ui/k^ = (1) + |ajj_/k^| (from (3.4.18)) = (1) , as N ^ XT 149 Lenmia 3.4.2 ; Under H (3.4.17) and the conditions A-D, |V.(Y.-3|X)| = Op(l) (3.4.20) as N ^ ", i = 1, . . . , k. Proof of lemma ; The result (3.4.20) is equation (3.22) in Sen (1969, p. 1675) , when using the notation of this section. Taking i = k, we note (3.4.20) yields: V. kl = l\(Ik-^k^H = °p(l) (3.4.21) as N ■> " . ■ Lemma 3.4.3 ; Let a^ be a positive constant and define I(aQ) = {a: |a| 1 a^ } . Then assuming conditions A-D, Vi(Ii-[Bi-a/N^a ]x) - V. (Y . - [ g, -b/N^a ]x) = p ('l',(j)) (a-b) [I*(F)] + o (1) (3.4.22) holds simultaneously for all a, b e I(a_) as N Proof of lemma : This is proven as Lemma 3.2 in Sen (1969, p. 1674). We have restated the lemma here in the notation of this section. 150 Remark ; Sen notes that since (3.4.22) holds simultaneously for all a, b e i (a^), it follows that (3.4.22) also holds if a and b are random variables depending on N and both are (1) as N ^ «. Lemma 3.4.4: Under H^ (3.4.17) and the conditions A-D, ^i " ^i^^i " ^k^^ = P('l',<(') [I*{F)]\^a^(6*-e*) + o (1) (3.4.23) as N ^ «>, i = 1, . . . , k. Proof of lemma: Vi = [V.(Y^-B*x) - V.(Y.-6|x)] + Vi(Y.-6^x). (3.4.24) Let a = N^a^(6.-6*) (3.4.25) and b = N^c:^(6.-6^). (3.4.26) By (3.4.19) and (3.4.18), a = (1) and b = O (1) as N > -. P p Then in view of the remark following the proof of Lemma 3.4.3 we may apply that lemma with a and b as in (3.4.25) and (3.4.26), respectively, obtaining 151 Vi(li-ejx) - v3_(Y^-ejx) = P (^F, (I)) [I*(F) ]\^a^(B^-6*) + o (1) (3.4.27) as N -> ", i = 1, ..., k. Apply (3.4.27) and (3.4.20) to (3.4.24) : Vi = [V^(Y.-e*x) - V^(Y.-B|x)] + V. (Y.-6^x) = [p(Y,*) [I*(F)]^N'^a^(e^-e*) + o (1)] + o (1) as N ^ 00, i = 1, _.^ k^ Thus (3.4.23) is verified. Lemma 3.4.5; Under H^ (3.4.17) and conditions A-D, ^''^x^^i-^k^ f N(kA.,[p2('F,,j,)I*(F)]-l) (3.4.28) as N -> =0, i = 1, ..., k. Proof of lemma ; By Lemma 3.4 in Sen (1969, p. 1676), N^cr^(e^-e.) f N(O,[p2(*,0)I*(F)]-^) (3.4.29) as N ->- ". Now under H„, — N' - ^T'2 '- 1?. 3^ N^a^(3*-e^-,./N^k^a^) 152 = N^a^(6|-6j^)-(a)^/k^) , In view of (3.4.29), this implie: %v d -1, N^a^(B|-Bj^) - (oj^/k^) ; N(0,[p''('^,<j,)I*(F)]"^) as N ■* ", and the desired result (3.4.28) follows immediately. Lemma 3.4.6: Let X X X = — n 2 ,n X k,n X = — n .o •l,n o 2,n • • . o k-l,n_ 1 ,n k,n 2 ,n k,n k-l,n k,n ' ^1 ^2 • p = • 1 ^k o u = '^l-^k ^2-^k ^k-l-^k (3.4.30) where X^, n >_ 1 represents a sequence of random vectors and _M is a vector of constants. Let a > be a constant. If Xn f ^k^ii'^'V (3.4.31) as n ->- <» , then '■*»»«^ »«W,— -: JSr -i^- - 153 2i° t Nk-i(l°'^^(Vi+Jk_i)) (3.4.32) as n -*■ °= . Proof of lemma; Consider an arbitrary linear combination of the elements of X°: — n k-1 ^ k-1 rO 2 ^^X. ^ = Z A. (X. -X, ) i=l 1 1'^ i=i 1 i.n k,n' k-1 k-1 = r x.x. ^ - X, z A. . ^^^ 1 i,n k'^i=i 1 Since X^ has a multivariate normal limiting distribution it follows (Serf ling, 1980, p. 18) that any linear combination ■ of the X^ must converge in distribution to the same linear combination of a random variable with that multivariate normal distribution. Hence ^-1 k-1 k-1 k-1 „ k-1 _ ,i,^i^i,n - ^k,n.i/i . N(.Z^X...-,^_Z_^A.,a2r(_z^.2 ) + k-1 ^ (.^ ^i) ]) (3.4.33) as n ^ °o . Letting A = k-1 (3.4,33) can be expressed as follows 154 A'^n t NU'y°,a2A'(Ik_i+Jk_i)i) (3.4.34) as n -> ". The result (3.4.32) follows (Serfling, 1980, p. 18), since (3.4.34) implies that any linear combination of the elements of X° converges in distribution to the same linear combination of a variable with the o 2 N. k_l (Ji 'Cr (Ik_i+J]^_2) ) distribution Before proceeding with the next lemma, we establish some additional notation. Let B^, g* , and co^, i = 1 , . . . , k, be as defined earlier and now define !* = '^ ^k 'k 'k 'k L J (3.4.35) 155 where ^^ has k elements. For a vector X v/ith k elements, let X be the vector of k-1 elem.ents formed by subtracting the kth element from each of the first k-1 elements of ^. This notation was used implicitly in the statement of Lemma 3.4.6. For example. 6* -B* ■^k-l ^k and 03 = '"l-'^k ' '^i "2-^k 0)2 « • • • • • '"k-r^k '^k-l ^ (3.4.36) since m, = . Lemma 3.4.7 ; Let _e*° and u° be as given in (3.4.36), and let -1 t^ = [p''('i',<^)I*(F)]"-'(Ij^_^+j^_^). (3.4.37) Under H (3.4.17) and conditions A-D , -N N' -O. d ^x<i* ) :Nk-i(^"^^ 'M (3.4.38) as N -> °° . Proof of lemma; Recall the definition of _6, and _B* in (3.4.35). From Lemma 3.4.5 and the independence of g* and ^"^ , , for i 7^ i ' , = »^ i .— ^1-..— — f.^--.. ., ,—— ^-- ^ ^■^- ..„ - - , — - ^ , , r - 156 it follows that N^^d*-!^) J Nj^{k-^,[p2(Y,^)i*(F)]-lT ) as N > CO, under E^. Now we identify N^a^{B*-3^) with X X — — K — n (3.4.30) and apply Lemma 3.4.6: N^x([l*-lk]°) t N(k-V,[P^(^,*)I*(F)](Ij^_^+J^_^)), * o „*o ^^k-l)) which, since [^ -_3 ] ° = 3 °, implie n'^^^CB*") f N{k-V.[P^^.<l>)IMF)](Ij^_^ as N ^ =0, and (3.4.38) has been proved. « For later use we note that (3.4.23) together with the i^ if ie fact that N'a^(B^-B^) has a normal limiting distribution (3.4.38) implies ^i = °p(l) (3.4.39) under H^ and conditions A-D, asN^oo, i = i^ ^ ]^_2 By Theorem 8.3.4 in Graybill (1969, p. 171), V^ = [P^(^,*)IMF)](I^_^-k-lj^_^). (3.4.40) 157 Leinma 3.4.8 : Under H (3.4.17) and conditions A-D, Na^(l*°)'lv^(l*°) f xJ_i(p2(^,q))I*(F)k-l Z i.,-Z}h z i=l (3.4.41) as N ->• ". Proof of lemma : From lemma 3.4.7 we have the following; •'V*°^''ic-i'x"V4v) as N -> ". ~k Let X be a random vector with the N,_ (k~''a) , | ) distribution. Then it follows (Serfling, 1980, p. 25) that Na X 2(B*°)'|;^(1*°) ^ X'tv^X (3.4.42) as N -> CO. The distribution of X'ty^X (Serfling, 1980, p. 128) is 2 ,1 -1 o ' x-l Ov Xk_l (k u ?v ii^ ) • (3.4.43) Evaluating the noncentrality parameter in (3.4.43), we have 158 0)° tyV = P^(^,<J>)I*{F) [a)°'lj^_^(.°-k"V'jj,_iiii°] i=l ^ i=l ^ k p^(Y,<t>)I*(F) E (co.-;;;)^, (3.4.44) i=l ^ which completes the proof of Leirnna 3.4.8. We now give the main result of this section, a theorem specifying the asymptotic distribution of the proposed test statistic L (3.4.16) under H,,. Theorem 3.4.9 ; Under H^^ (3.4.17) and conditions A-D, L t Xk_i(P^(Y.<!>)IMF)k"\z {i^^-Z)^) (3.4.45) k as N -> 00, where L is given by (3.4.16) and oj = Z cj./k. i=l ^ Proof: k — I L = Z (V.-V)^ i i=l ^ = [ z vl'] - [k -^( Z V.)^] i=l ^ i=l ^ 159 i=l ^ ^ i=l ^ i=l ^ k = Z V^ - k ( Z V )^ + V^ - 2k~-^V, Z v.. (3.4.46] i=l i=l ^ ^ ^i=l 1 By Lemma 3.4.2 (see (3.4.21)) l\I = Op(l) (3.4.47) as N > ", and in (3.4.39) we noted IV^I = Op(l) (3.4.48) as N ^ 06, i = 1^ — ^ ]^-l^ Consequently, applying these results to (3.4.46) yields k-1 k-1. L = Z V - k ( Z V )^ + o (1) (3.4.49) i=l ^ i=l ^ P as N ^ 00. By Slutzky's Theorem (Randies and Wolfe, 1979, p. 424) it follows from (3.4.49) that if L has a limiting distribution as N ^ oo, it is the same as the limiting distribution of k-1 k-1. L^ = Z V - k~'( Z V )^ (3.4.50) 1=1 i=l 160 Leirana 3.4.4 showed the following: Vj, = p(¥,ct,) [lMF)]\^a^(iB|-B*) + Op(l) (3.4.51) asN->", i = l, ...,k. Once again appealing to Slutzky's Theorem, it follows from (3.4.51) and the continuity of the quadratic form (3.4.50), that the limiting distribution of L^ (and hence of L) is the same as the limiting distribution of 2 2 ^"^ 9 1 k-1 o p ('J',f)I*(F)Na^[ Z (e:f-6*)^ - k~-^( E (6*-3*))^]. ^ 1=1 ^ ^ i=l ^ ^ (3.4.52) Using the matrix notation _3*° (3.4.36) and t^^ (3.4.40), the expression (3.4.52) is equal to Na;(l*°)'t;^l*°). (3.4.53) From Lemma 3.4.8, (3.4.53) has a limiting (N^") 2 2 -1 ^ ? Xj^_l(p {'V,<p)I*{F)k Z (o) -to)'') distribution so (3.4.45) i=l holds and the proof is complete. The asymptotic distribution of L (3.4.16) under 2 ^0" ^1 = ^2 " • • • " ^k ^^ ^k-l^°^' "^^^^ follows from Theorem 3.4.9 if w^ = 03^ = ... = oi^ = . As used in (3.2.77), let ^y^_^ ^ be the upper 100 (1-a) percentile of the 161 Xj^_-[^(0) distribution. Then the test of H against H (3.4.1) that rejects H when ^ - ^k-l,a (3.4.54) is an approximate level a test. Comparisons of the test (3.2.77) based on S (3.2.74) with this test will be made in the next section by calculating the tests' asymptotic relative efficiency. 3.5 Asymptotic Relative Efficiencies In Sections 3.1 and 3 . 2 we proposed and developed a test comparing the slopes of several regression lines with a standard or control. A competitor to this proposed test was suggested in Section 3.4. We now compare these two tests by examining the Pitman asymptotic relative efficiency (PARE) of the corresponding test statistics. Recall in Section 3.2, assuming model (3.1.1), the proposed test of the null hypothesis -0* ^1 " ^2 " ••• " ^k' (3.5.1) against the alternative, -1" '^i ^ ^k ^°^ ^^ least one i, (3.5.2) 162 was based on S - T't ^T, (3.5.3) where T is a k-1 dimensional vector with elements T^(0) = (Na^^^)"^ i:Zx^gSgn(Z^g-Z^^), (3.5.4) i = 1, ..., k-1. The Statistic T.(0) compares the slope of the ith line with the slope of the kth line. Statistics having the form of T. (0) were proposed and discussed at length in Chapter Two, when comparing the slopes of two lines. The matrix |~ appearing in (3.5.3) is the inverse of the asymptotic covariance matrix of T (see (3.2.73)). In the previous section, we introduced a second statistic for testing H , based on a statistic proposed by Sen (1969). Again assuming model (3.1.1), recall the statistics V^ = [ Z (x.-x)Ej^ ]/(AN'a ), (3.5.5) j = l -■ ij i = 1, ..., k, where R . . is the rank of Y. . among the observations of the ith line, E_ are the scores (3.4.4) or ID (3.4.5), and A is the constant in (3.4.7). Let Bf, denote the Hodges-Lehmann estimate based on V, and let 163 Vi = V.(Y.-3*x) (3.5.6) equal the value of V. when using Y. - B^x^ , Y. - B*x ••" '^iN ~ ^k^N' ^^^^ ^^^ statistic based on Sen's work is ^ k - L = Z (V.-V)^ (3.5.7) i=l where V = Z V./k. i=l ^ Let co^, 012, ..., (jjj^ be given constants with u = 0. For use in the next result define -1 '^ _ p ^2 _ _i=l_ ^w =2 ' (3.5.8) 0) -1 ^ where w = k E oj . . We derived the asymptotic distribu- i=l ^ tions of S and L under the sequence of alternatives. H^: ^i = ^k ^ a)^/N^k^a^, (3.5.9) -L -1-f •••^ jC. 164 These asymptotic distributions will be used to arrive at the PARE of the two statistics. Recall the test (3.2.77) based on the generalization of the Sievers-Scholz statistic, S (3.5.3), depends on the underlying error distribution F through A* = 12A (F) . In practice we suggested a conservative version of the test (3.2.77) based on replacing A with A given by (3.2.84). However for asymptotic efficiency comparisons investigating the relative merits of the Sievers-Scholz and Sen methods, it is appropriate to use the test (3.2.77) based on S. The resulting PARE expressions will be evaluated assuming different specific error distributions. Result 3.5.1; Let c^, 1(g), a*, p(^,(j)), and I* (F) be as defined in (3.5.8), (3.2.13), (3.2.12), (3.4.11), and (3.4.9), respectively. Assuming conditions I-IV and A-D, the PARE of S (3.5.3) with respect to L (3.5.7) under H — N (3.5.9) is ^2 * PARE(S,L) = ^^^ W) [1 + 1 - 2A (l-A*)p^(Y,*)I*(F) (l+(k-2)A*)c^ 0) (3.5.10) Verification of Result 3.5.1 ; Under H^^ and the assumed conditions. Theorems 3.2.8 and 3.4.9 give the limiting distributions of S and L, respectively, as follows: 165 k-1 ^ k-1 ^ c (l+(k-2)A*) Z cor-A*{ E 0).) (1-A*) (l+(k-2) A*) (3.5.11) k L f X? 1 (p^(^,<j>)I*(F)k~^ E (to.-u)^). (3.5.12) ^ ^ i=l 1 VJith some algebraic manipulations, the noncentrality parameter for S in (3.5.11) can be expressed as follows; 2 k * 121 (g) .,-1 „ . -x2,r. 1 - 2A ,- _ -^, * t^ ^ ((i^--w) J [1 + J 5"]. (3.5.13) 1-A i=l -^ (l+(k-2)A )c CO Since both S and L have limiting noncentral chi-squared distributions under H with equal degrees of freedom, it follows, as discussed in the verification of Result 2.4.5, that the PARE(S,L) is given by the ratio of the noncentrality parameters. Then using the noncentrality parameters for S and L given in (3.5.13) and (3.5.12), respectively, the PARE (3.5.10) follows imm.ediately . The PARE(S,L) (3.5.10) depends on the constants 2 ,, w-f '•'/ u, through c (3.5.8). Hence we give bounds for this PARE that are independent of the to ' s . We first state and prove a needed lemma. tii Lemma 3.5.2 : For arbitrary constants oi^ , CJ, ,^, . . . , ^^ satisfying w, = and m . ^ for some i = 1, ..., k-1, define 166 1 0). = (1/i) s u and (3.5.14) 2 ^ - 2 ^i - .^ ((Jj-o)^) (3.5.15) for i = k-1, k. Then 1,-2 V -2~ - ^~^- (3.5.16) ^k Proof of lemma; First note ""k-l = (k/k-l)^;;;^. (3.5.17) Then = 2 _ ^ 2 , -2 ^ i=l ^ k ,E^ .2 _ (3,_i,-2_^ ^ (k-l)^2_^ _ ^-2 ^k-1 ^ ^^ /k-l)a)^_i - k[;;J (from (3.5.17)) 167 = sj_^ + kmj[(k/(k-l)) - 1] ^k-1 + [k;;;^/(k-i)] , -2 k —2 = (k-l)[l - (sj_^/sj)]. ■ (3.5.18) ^k The right hand side of (3.5.18) is maximized when s"^ =0 k-1 ' in which case (3.5.16) follows immediately. Result 3.5.3: The PARE (S,L) given by (3.5.10) has the following bounds: 121 (g) * 1 ot2 / ^ ■* 2~^ * 1 P^^^ (S,L) < — ^^^ ^?/ (1-A )p" (¥,<(,) I'" (F) ■ ■ - A*p2('^,<^)l*(F) (3.5.19) Verification of Result 3.5.3 : Recall the bounds for A = 12A (F) , 1/3 < A < 1/2, applicable under continuous F. Then since 1-2 A* > , it follows that * 1 - 2A * 2 - ° (3.5.20) (l+(k-2)A )c 0) ■"a'MB>Btww'»iair3i^— f— — niiuM n 168 and so the lower bound in (3.5.19) is obtained by applying (3.5.20) to the PARE (3.5.10). To obtain the upper bound, first note that (3.5.16) of Lemma 3.5.2 can be expressed as — 1 k-1. (3.5.21) c Applying (3.5.21) to the PARE (3.5.10) yields 2 PARE (S,£) < ^-i2I__(|)_ ^ k ^^ ^ (3.5.22) p (Y,(j))I (F) l+(k-2)A Then since A* < 1/2 => ^ ^ -X_, (3.5.23) l+(k-2)A A the upper bound in (3.5.19) is obtained by applying (3.5.23) to (3.5.22). Suppose we assume Wilcoxon scores, 4)(u)=u, 0<u<l. Then, as derived in Chapter Two, it follows from (2.4.34), (2.4.35) , and (2.4.39) that P^('i',*)I*(F) = 12l^(f) . (3.5.24) Hence we substitute (3.5.24) into (3.5.19) resulting in t" 'Ifll.BHI' 'lipr^'ir ■ iiil'l j-llMLli.ili 169 i^(g) (l-A*)I^(f) < PARE (S,L) < t2. £L — * 2 ' A I^(f) (3.5.25) assuming <i> (u) = u. Since A = 12A (F) , values of A may be obtained from Table 2 4 for several common error distribu- tions. Table 25 gives the upper and lower bounds of the PARE(S,L) for four error distributions, assuming (p (u) = u. We see that the PARE (S,L) is close to one for the three distributions with light to moderately heavy tails (uniform, normal, double exponential) , but has value close to 1/2 under the heavily tailed Cauchy distribution. These results are essentially the same as the PARE comparisons of the Sievers-Scholz and Sen statistics in Table 2 for the two line setting. Thus, as remarked at the beginning of this chapter, the superiority of Sen's test with respect to Table 25. Upper and Lower Bounds of PARE(S,L) for Selected Error Distributions Assuming (u) = u. Distribution Uniform Normal Double Exponential Cauchy Lower Bound Upper Bound 0.87 0.97 0.74 0.46 0.91 1.04 0.83 0.55 170 the Sievers-Scholz test in terms of PARE is not surprising, given our results in Chapter Two and the fact that Sen's test maximizes the efficiency relative to the likelihood ratio test. Note, however, that assuming (f) (u) = u, the PARE of Sen's test to the classical least squares theory test under double exponential errors is 1.50 (Sen, 1969, p. 1676; Hollander and Wolfe, 1973, p. 64) and the corresponding bounds of the PARE of the Sievers-Scholz test based on S to the classical test are 1.11 and 1.24. The point here is that although the PARE (S,L) favors Sen's test under heavily tailed distributions, the difference between the tests' PARES with respect to the least squares theory test is not impressive. Also, recall from Section 3.3 that the statistic S has an advantage over L since the form of S s allows the construction of exact conditional ta'ts of H. that are computationally feasible under small sample sizes. The iterative computations necessary to perform Sen's test preclude the possibility of computationally feasible, exact tests based on his statistic. Furthermore, simulation studies by Lo, Simkin, and Worthley (1978) indicate the power of Sen's test with respect to the classical least squares test in the case of three regression lines under small samples is quite conservative. In view of these points, the proposed generalization of the Sievers-Scholz method to the setting of several regression lines has merit as a robust, computationally simple technique allowing exact tests that are feasible under small sample sizes. g*TT * —t tm-- i CHAPTER FOUR CONCLUSIONS We now summarize the conclusions resulting from the work in Chapter Two and Chapter Three. In Chapter Two we considered the case of two regression lines. Assuming common regression constants for the two lines, we proposed the application of a statistic due to Sievers and Scholz (2.1.2) to the observed differences Z., j=l, ..., n, given by (2.1.3). The proposed statistic (2.1.2) employs weights, ^rs' ^^^^ ^^® chosen by the user'. If the regression con- stants are x^, , x^^, then when using the weights ^^2 ~ ^s ~ ^r' ^^^ null distribution of the proposed statis- tic depends on the regression constants. An associated exact confidence interval for the slope difference can be obtained by calculation of a permutation distribution. We found this to be feasible with the use of a computer when the number of regression constants was small, say less than eight per line. When using the weights a = 1 if x > x . rs s r' zero otherwise, the proposed test statistic essentially reduces to a statistic due to Theil and Sen (2.1.4), com- puted using the observed differences Z . , j = 1 , . . . , n. In this case, the null distribution of the proposed statistic depends on Kendall's tau. An associated exact confidence 171 172 interval for the slope difference can be calculated using readily available tabled critical values of this distribu- tion. VJe proposed this procedure when use of the weights a = X - X was not feasible, rs s r Pitman asymptotic relative efficiencies (PAREs) of these two proposed tests with respect to the tests of Hollander, Rao and Gore, Sen, and the classical t-test were computed assuming equal spacing of the regression constants. Choice of the weights a = x - x or the zero-one weights ^ rs s r ^ does not affect the PARE under equal spacing, so results are given jointly for these two methods. The PARE of the proposed method relative to Hollander's method is greater than one over a wide variety of error distributions. The PARE of the proposed method with respect to the Rao-Gore technique is 4/3 irrespective of the underlying error distribution (subject to certain regularity conditions) . Simulation results assuming unequal spacing of the regres- sion constants also favor the proposed methods over the other two exact procedures (the Hollander and Rao-Gore methods) . The PARE of Sen's (1969) test with respect to the proposed test is greater than or equal to one under most common error distributions. However, Sen's approach is only asymptotically distribution-free and requires iterative techniques. The PAREs and simulation results showed the classical t-test performs better than the proposed methods under distributions with light tails. However, the 173 classical test does very poorly when the underlying error distribution has heavy tails, such as the Cauchy distribu- tion. For these reasons we feel the proposed methods are preferred when exact, distribution-free procedures are desired to make inference about the slope difference, assuming common regression constants are used for both lines. In Chapter Three we considered the case of several regression lines. Assuming common regression constants, we defined a set of statistics, each having the form of the proposed statistic of Chapter Two. Specifically, when comparing the slopes of k lines, k-1 statistics were con- sidered. The ith statistic compared the slope of the ith and kth lines. We proposed a test based on a quadratic form (3.2.74) in this set of k-1 statistics. Hence this test was designed to detect alternatives where one or more of the slopes of the first k-1 lines differ from the kth line. This is the case where the kth line is considered to be a standard, or control. A modification of Sen's (1969) test was constructed as a potential competitor to the proposed test. Loxver and upper bounds of the PAREs of the proposed test with respect to the modification of Sen's test were calculated for selected error distributions (see Table 25) . Although these bounds favor Sen's test under the very heavily tailed Cauchy distribution, they are close to one for the uniform, normal, and double exponential distributions. We showed that exact 174 conditional tests based on the proposed statistic are computationally feasible under small sample sizes, a feature not shared by Sen's statistic. Hence the method proposed in the setting of several regression lines has merit as a relatively robust, computationally simple technique allowing exact tests. BIBLIOGRAPHY Adichie, J.N. (1967) . Estimates of Regression Parameters Based on Rank Tests. Ann. Math" Stat. , 38, 894-904. Adichie, J.N. (1974). Rank Score Comparison of Several Regression Parameters. Ann, of Stat. , 2, 396-402. Boyett, J.M., and Shuster, J.J. (1977). Nonparametric One-sided Tests in Multivariate Analysis with Medical Applications. J. Amer. Stat. Assoc , 72, 665-668. Chow, Y.S., and Teicher, H. (1978). Probability Theory; Independence, Interchangeability , Martingales . New York: Springer-Verlag. Crow, E.L., and Siddigui, M.M. (1967). Robust Estimation of Location. J. Amer. Stat. Assoc , 62, 353-389. Dwass, M. (1957). Modified Randomization Tests for Non- parametric Hypotheses. Ann. Math. Stat. , 28, 181-187. Finney, D.J. (1964) . Statistical Method in Biological Assay . London: Charles Griffin and Company. Graybill, F.A. (1969) . Introduction to Matrices with Applications in Statistics . Belmont, California: Wadsworth. Hajek, J. (1962) . Asymptotically Most Powerful Rank Order Tests. Ann. Math. Stat. , 33, 1124-1147. Hajek, J., and Sidak, Z.S. (1967). Theory of Rank Tests . New York: Academic Press. Hodges, J.L., and Lehmann, E.L. (1963). Estimates of Location Based on Rank Tests. Ann. Math. Stat., 34, 598-611. Hoeffding, W. (1951) . Optimum Nonparametric Tests. Proc Second Berkely Symp. Math. Stat. Prob. , 1, 82-92. Hollander, M. (1970) . A Distribution-Free Test for Parallelism. J. Amer. Stat. Assoc , 65, 387-394. 175 «^f.*r'».-Tu;:=: 176 Hollander, M. , and Wolfe, D.A. (1973). Nonparametric Statistical Methods . Nev/ York: John Wiley and Sons. Ireson, M.J. (1983) . Nonparametric Regression in the Analysis of Survival Data. Ph.D. Dissertation in Statistics, University of Florida. Jaeckel, L.A. (1972) . Estimating Regression Coefficients by Minimizing the Dispersion of the Residuals. Ann. Math. Stat. , 43, 1449-1458. Jureckova, J. (1971) . Nonparametric Estimate of Regression Coefficients. Ann. Math. Stat. , 42, 1328-1338. Kendall, M.G. (1955) . Rank Correlation Methods . London: Charles Griffin and Company. Knuth, D.E. (1973) . The Art of Computer Programming . Reading, Mass.: Addison-Wesley . Lehmann, E.L. (1963) . Nonparametric Confidence Intervals for a Shift Parameter. Ann. Math . Stat., 35, 1507- 1512. Lo, L.C., Simkin, M.G., and Worthley, R.G. (1978). A Small-Sample Comparison of Rank Score Tests for Parallelism of Several Regression Lines. J. Amer. Stat. Assoc. , 73, 666-669. Mann, B.L., and Pirie, W.R. (1982). Tighter Bounds and Simplified Estimation for Moments of Some Rank Statis- tics. Commun. Stat. — Theory an d Methods, 11, 1107- 1117. "" Mehta, C.R., and Patel, N.R. (1983). A Network Algorithm for Performing Fisher's Exact Test in r x c Contingency Tables. J. Amer. Stat. Assoc , 78, 427-434. Noether, G.E. (1949). On a Theorem bv Wald and Wolfowitz. Ann. Math. Stat. , 20, 455-458. Noether, G.E. (1955) . On a Theorem of Pitman. Ann. Math. Stat. , 26, 64-68. Pagano, M. , and Tritchler, D. (1983). On Obtaining Permutation Distributions in Polynomial Time. J. Amer. Stat. Assoc. , 78, 435-440. Randies, R.H., and Wolfe, D.A. (1979). Introduction to the Theory of Nonparametric Statistics . New York: John Wiley and Sons. 177 Rao, K.S.M., and Gore, A. P. (1981). Distribution-Free Tests for Parallelism and Concurrence in Tv/o-Sample Regression Problem. J. Stat. Plannina and Inference, 5, 281-28 6. Scholz, F.W. (1977). Weighted Median Regression Estimates, Inst. Math. Stat. Bulletin , 6, 44. Sen, P.K. (1966) . On a Distribution-Free Method of Esti- mating Asymptotic Efficiency of a Class of Nonpara- metric Tests. Ann. Math. Stat. , 37, 1759-1770. Sen, P.K. (1968) . Estimates of the Regression Coefficient Based on Kendall's Tau. J. Amer. Stat. Assoc, 63, 1379-1389. Sen, P.K. (1969) . On a Class of Rank Order Tests for the Parallelism of Several Regression Lines. Ann. Math. Stat. , 40, 1668-1683. Serfling, R.J. (1980) . Approximation Theorems of Mathe- matical Statistics . New York: John Wiley and Sons. Sievers, G.L. (1978). Weighted Rank Statistics for Simple Linear Regression. J. Amer. Stat. Assoc , 73, 628-631. Smit, C.F. (1979) . An Empirical Comparison of Several Tests for Parallelism of Regression Lines. Commun . Stat. — Simula. Computa. , 8, 61-74. Terry, M.E. (1952) . Some Rank Order Tests Which are Most Powerful Against Specific Parametric Alternatives. Ann. Math. Stat. , 23, 346-366. Theil, H. (1950) . A Rank-Invariant Method of Linear and Polynominal Regression Analysis. I, II, III, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings , 53, 386-392, 521-525, 1397-1412. BIOGRAPHICAL SKETCH Raymond Richard Daley was born in West Palm Beach, Florida, on March 3, 1957. He moved to Titusville, Florida, in 1962. After graduating from Titusville High School in 1975, he enrolled at the University of Central Florida. He received his Bachelor of Science degree in statistics in 1978 and was honored at his commencement for having attained the highest grade point average of his graduating class. He entered Graduate School at the University of Florida in 1978 and later received his Master of Statistics degree in 1980. He expects to receive the degree of Doctor of Philosophy in December, 1984. His professional career has included work as a statis- tician for an Air Force climatology study at the University of Central Florida and consulting in the Biostatistics Unit of the J. Hillis Miller Health Center at the University of Florida. He has received Graduate School fellowships and graduate assistantships during his academic career at the University of Florida. He is a member of the American Statistical Association and the Biometric Society. 178 _ _I certify that I have read this study and that in my opmxon It conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophv. \ Pen aver V. Rao, Chairman Professor of Statistics I certify that I have read this study and that in my opinion It conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. U M d?a. Ronald H. Randies Professor of Statistics I certify that I have read this study opinion it conforms to acceptable standa presentation and is fully adequate in sc a dissertation for the degree of Doctor o and that in my s of scholarly and quality, as f Philosophy. rd ope C^ . CT^ ^ JohnjGp Saw Professor of Statistics